Let's build a universe.
To describe this universe, we need a metric. I won't go into details about the precise definition - for more, see Wikipedia, as well as Physics and Mathematics. In this case, we need a metric of dimension (4 + 2) (i.e. four dimensions of space and two of time). This will be represented in a 6-by-6 matrix:
$$g=\begin{bmatrix} g_{t_1t_1} & 0 & 0 & 0 & 0 & 0 \\ 0 & g_{t_2t_2} & 0 & 0 & 0 & 0 \\ 0 & 0 & g_{rr} & 0 & 0 & 0 \\ 0 & 0 & 0 & g_{\theta\theta} & 0 & 0 \\ 0 & 0 & 0 & 0 & g_{\phi\phi} & 0 \\ 0 & 0 & 0 & 0 & 0 & g_{\varphi\varphi} \end{bmatrix}$$
This corresponds to a line element1 of
$$ds^2 = - dt_1^2 - dt_2^2 + dr^2 + r^2 (d\theta^2+ \sin^2\theta (d\phi^2+\sin^2 \phi d\varphi^2))$$
using n-spherical coordinates and a metric signature of (-,-,+,+,+,+), which happens to be my personal preference. You can use (+,+,-,-,-,-), if you want.
That describes an empty, four-dimensional Riemannian manifold in 4-dimensional n-spherical coordinates. It's your universe. The problem is, there's nothing in it. For that, we turn to the Friedmann–Lemaître–Robertson–Walker (FLRW) metric.
One of the beautiful things about the FLRW metric is that it's an exact, homogenous, isotropic perfect fluid solution of the Einstein Field Equations that can be used for modeling universes. Another beautiful thing is that it's so simple, compared to some of the other wacky stuff that you can get out of the EFEs. A third beautiful thing is that the FLRW metric yields the Friedmann equations. But I'll get to that a bit later.
Thank you so much for saying that the two time dimensions can act as one. This simplifies things greatly, because it means that they behave identically. In other words, $g_{t_1t_1}$ is the same as $g_{t_2t_2}$. It's great.
Now the FLRW metric includes a scale factor, $a(t)$, which is a function of time and describes the expansion or contraction of the universe (or neither, if $a(t)=1$). The reason that it's good that the two time dimensions are the same is because if they weren't, I'd have to write the scale factor as $a(t_1,t_2)$. That's not great, because the Friedmann equations involve derivatives of $a(t)$. Partial derivatives would make things more complicated. Anyway, on to the Friedmann equations.
There are just two:
$$\frac{\dot{a}^2+kc^2}{a^2}=\frac{8 \pi G \rho + \Lambda c^2}{3}$$
and
$$\frac{\ddot{a}}{a}=-\frac{4 \pi G}{3} \left(\rho + \frac{3p}{c^2} \right) + \frac{\Lambda c^2}{3}$$
The meanings of the variables are given nicely on Wikipedia.
These two equations describe how the universe expands or contracts. They can tell us quite a lot about its behavior. If you want to make the universe interesting, do try it. But if you want a static universe, then $a=1$, and all derivatives are $0$.
The FLRW metric is a perfect fluid solution, which is quite handy here. Wikipedia again gives an interesting relation. Take a fluid with pressure $p$ and density $\rho$. The equation of state is
$$p=w \rho c^2$$
We can then write $a(t)$ as (through an irrelevant derivation)
$$a(t)=a_0 t^{\frac{2}{3(w+1)}}$$
So solve for $w$ above, and you can find your scale factor. This means, actually that your universe may be expanding or contracting. From here, we can calculate all sorts of cool things.
With a scale factor of $a \neq 1$, the FLRW metric is
$$ds^2 = - dt_1^2 - dt_2^2 + a(t)(dr^2 + r^2 (d\theta^2+ (\sin^2\theta d\phi^2+\sin^2 \phi d\varphi^2)))$$
and, substituting in for $a(t)$,
$$ds^2 = - dt_1^2 - dt_2^2 + a_0 t^{\frac{2}{3(w+1)}}(dr^2 + r^2 (d\theta^2+ (\sin^2\theta d\phi^2+\sin^2 \phi d\varphi^2)))$$
That's your universe.
We have to figure out a plausible way for the bubbles and tunnels to form. In our universe, gas clouds form because of gravity. Then stars form. Here, though, gravity would be trying to collapse the matter around the empty spaces.
You could go for something like Dan's "Harmonics", or you could treat the "vacuum" as another fluid. In fact, you have too, because you have to explain why the fluid around these pockets doesn't collapse. Here's an example.
Take a gas cloud. This cloud has a few quantities: temperature ($T$), pressure ($p$) and density ($\rho$), as well as maybe a few other non-vital characteristics.2 Changes in one will influence changes in the other two. To start with, though, these properties are constant.
Now, the cloud has pressure. This means that every bit of it pushes against regions of space nearby it. This, along with the other bits, means that unless gravity is strong enough to keep the cloud in hydrostatic equilibrium - or, in fact, to make it undergo gravitational collapse - the cloud may expand outwards.
In this case, the situation is that of a cavity inside a gas cloud. There's nothing to stop the gas from moving in, whereas there is pressure from the gas in the cloud. These openings will be crushed very quickly. So you need these bubbles to be more like a fluid that the absence of one.
Another thing to add is that the fluid in this universe should be more or less uniform. For example, in our universe, any bit of vacuum is, in general, much the same as any other bit of vacuum. Why should this be any different in this universe? What makes any bit of this fluid special when compared to any other bit? You need some processes to occur to destroy the local homogeneity and isotropy of the fluid. On large scales, though, these properties can not be violated - otherwise the conditions for the FLRW metric are not met.
I'd like to go on with a discussion of a passage from Wikipedia:
In 1920, Paul Ehrenfest showed that if there is only one time dimension and greater than three spatial dimensions, the orbit of a planet about its Sun cannot remain stable. The same is true of a star's orbit around the center of its galaxy. Ehrenfest also showed that if there are an even number of spatial dimensions, then the different parts of a wave impulse will travel at different speeds. If there are 5 + 2k spatial dimensions, where k is a whole number, then wave impulses become distorted. In 1922, Hermann Weyl showed that Maxwell's theory of electromagnetism works only with three dimensions of space and one of time. Finally, Tangherlini showed in 1963 that when there are more than three spatial dimensions, electron orbitals around nuclei cannot be stable; electrons would either fall into the nucleus or disperse.
Max Tegmark expands on the preceding argument in the following anthropic manner. If T differs from 1, the behavior of physical systems could not be predicted reliably from knowledge of the relevant partial differential equations. In such a universe, intelligent life capable of manipulating technology could not emerge. Moreover, if T > 1, Tegmark maintains that protons and electrons would be unstable and could decay into particles having greater mass than themselves (This is not a problem if the particles have a sufficiently low temperature).
Translation:
Ehrenfest's work has one huge consequence: orbital mechanics are not fun. Stable orbits are out the window. This means no planetary systems whatsoever - and given the nebular hypothesis, this means that it would be tough for planets to even form! So scratch and life like we know it - although I take it you already expected that. Your description is of something wildly different from our universe - extremely chaotic. I think you expected that.
Weyl's work (available here, if you're willing to trudge through it, which I'm currently not) chucks electromagnetism as we know it out the window. I have yet to find the relevant section, so it's unclear if electromagnetism cannot exist in any form or whether Maxwell's formulation is merely inadequate. Let's just not expect anything special.
Tegmark's work may be the most relevant (though I'm skeptical of some of his other work - I won't take that against him). If he's correct, then your two time dimensions must behave exactly as one. Or you can change up your elementary particles. Otherwise, no stable atoms. This answers the "How would the different elements occur, be synthesised and/or destroyed?" bit. They wouldn't. Add to that the bit about Tangherlini - whose work I'm not familiar with - and you can kiss elements goodbye.
On to what your universe will actually be like.
I'll start at the smallest scale: elementary particles.
If Tegmark is right, electrons and protons (and therefore, most likely, quarks) must go out the window. The same goes for the electron's heavier cousins, the muon and the tau. Quarks take away all the baryons (composite particles made up of quarks), leaving us with neutrinos. I don't know if those, too, may not exist. If so, their interactions would be very dull! Neutrinos seldom interact with anything.
We could come up with a whole new set of elementary particles, or we could use a little loophole. See, according to the Freidmann equations (and their variants), if $a(t) \propto t^{2/3}$, then the universe is dominated by matter. If $a(t) \propto t^{1/2}$, then the universe is dominated by radiation.
Hearken back to the work around $p=w \rho c^2$, and the later derivation. In that, the scale factor is
$$a(t)=a_0 t^{\frac{2}{3(w+1)}}=a_0 t^{\frac{2}{3w+3}}$$
For the universe to be radiation dominated, the thing that $t$ is raised to must be proportional to $1/2$. An easy way of doing this is to set $w$ to $1/3$. We then find that $t$ is, in fact, raised to the power of $1/2$, and so our universe is radiation dominated.3
Note that our universe entered a stage like this very early in its history. Its primary constituents? Photons and neutrinos. Sounds a bit like ours. The cool thing? When our universe was about 378,000 years old, it underwent recombination. Up until that point, it was filled with an opaque (to light) plasma. That was when the CMB was formed.
At this point, it's looking a lot like our universe in its earliest days.
On to some of your specific points.
What set of physical laws could achieve this universe?
You can set virtually any laws you want. For example, the equation of state used above doesn't have to be accurate. Maybe there's another constant shoved in there. Maybe general relativity doesn't work, and so the FLRW metric doesn't accurately describe the universe. Take your pick. You can create new constants for this universe, and by doing so create new laws and interactions. The number of dimensions doesn't matter.
You do have to be careful about making sure that the laws lead to the desired scenario. For example, if you were to introduce a force that pushes matter away - sort of like anti-gravity - then your scenario doesn't make sense. Everything is trying to get away from everything else. All of a sudden, pressure skyrockets, and things start to get weird. Weirder than you intended, that is.
If you end up coming up with some basic laws, then I can get back to you on what else would happen.
In addition, to fill the requirements of the strong anthropic principle (this need not be simulatable), where does light and heat come from?
I don't know about you, but I would not want to live here. Life as we know it could not survive here. Photons and neutrinos (well, their equivalents in this universe) aren't the best food.
Fortunately, you didn't ask about life; you asked about light and heat. Those are easier. The universe is filled with radiation, although if it hasn't undergone recombination, it may be opaque to photons. But it's quite possible that, in some places, photons can flow freely, spreading heat with them. Shine light on anything and it will heat up, and vice versa.
Not a very hospitable place, I'll grant you.
How would day/night or light/dark cycles occur?
You need to figure out just how the heck you have a planetary system form. Before, I discussed how that would not be possible. Maybe an isolated ball of rock could form. But that would be difficult. It would have a nice sky, though. Maybe reminiscent of Dave Bowman's journey in 2001: A Space Odyssey.
Same goes for
How could 'earth'-quakes occur?
You can't have earthquakes without Earth.
How would the different elements occur, be synthesised and/or destroyed?
I'll end on a positive note, because, as usual, it seems I've inadvertently written a pessimistic answer. Damn. I'll add to it by recalling the work of Tangherlini, which says that elements will not form. Shoot.
But here's the positive note: You can create elementary particles that circumvent that (somehow). Science can be a real pain in the neck sometimes, but you have to remember that you are always in control. In a world like this, you can create whatever you want. Remember the metric? That gives one description of the universe. One. And look at all the variables that can change. I barely used any actual numbers in this answer. They're there for you to fill in.
Create your own laws. Your own particles. Your own universe. And you've got a universe that I'd be happy to live in, because it came from unbounded imagination. That's pretty awesome.
1 For more information on this particular line element, see Wolfram Math World.
2 We can, by the way, related these properties via the ideal gas law, $PV=nRT$. You just need to derive density from the law.
3 And so $p= \frac{1}{3} \rho c^2$.
Second Try
I apologize for the potentially confusing second answer, but at the moment, my original one is rather long-winded, and I'd like to go in a whole new direction from it. To avoid confusion, I'll start anew. If people want, I can delete this and merge it1 into the other answer, but for simplicity and readability I'd like to give it its own section
Correct me if I'm wrong, but it seems that you want a set of rules that, if given at the very start, could predict how your universe would evolve, both on large and small scales (though perhaps more on small scales). In other words, a deterministic universe that could be simulated on a computer (I'm not suggesting a relation to any of the simulated-world questions).
Properties
Matter has to have some properties that dictate how it interacts with other bits of matter. Not all bits of matter are influenced by the same things: For example, in our universe, some particles have electric charge while others do not. Yet we can still describe them in terms of electric charge: $q=0$. So I'll say that these properties - represented as variables, though some are variables and some are constants for a given particle - can apply across-the-board.
- Gravitational mass, $m$: This is a universe where active and passive gravitational mass are the same, though inertial mass may not be.
- Inertial mass, $m_i$: I'd like to keep this equivalent to gravitational mass, but it may not be the case. The choice is yours.
- Position, coordinate system of your choice: I'd use spherical (hyperspherical, in this case) coordinates globally, as I discussed in my original answer, but you can use $x,y,z$-coordinates in any scenario, if you want. It's simple for discussing two-dimensional interactions.
- Speed/velocity, $v$: Particles can move; therefore, they have a velocity. From this and mass we can say that they have momentum ($p$) and kinetic energy ($KE$). Higher time derivatives also exist (the two time dimensions behave as one), so particles also have acceleration ($\dot{v}=a$), jerk ($\dot{a}=j$), etc.
- Temperature, $T$: Particles can vibrate; therefore, they have a temperature.
Properties not in our universe2:
- Ikimgiir charge, $i$: A property describing how likely a particle is to interact with other particles via the ikimgiir force, as described later. Ikimgiir charge ranges from $0$ to $1$.
- Kaaziikkhaaku, $k$: A property describing how a particle may attach to another particle of the same type and form a composite particle, as described later.
There are also other properties (pressure, density, etc.) that can be derived on large scales. In fact, velocity and high time derivatives of position are really also derived quantities.
Forces:
- Gravity: Gravity works the same here as in our universe, and can be described by general relativity (albeit by a 4 + 2 metric). It's simpler, though, to use a classical approximation on smaller scales. So our law here is
$$F=G\frac{M_1m_2}{r^3}$$
with $G$ being any value you choose. It falls off with $r^2$ and not $r^2$ because of a generalization of the inverse-square law to higher dimensions.
Ikimgiir: Ikimgiir describes the process of forming those cavities you mentioned. I said that the charge ranges from $0$ to $1$. That's because it's similar to a probabilistic measure. The charge is measured across all particles as a normal distribution: There are fewer particles with low ikimgiir charges and fewer particles with higher ikimggir charges. This distribution, though, can vary across types of particles. For example, particle type A might have a different distribution than particle type B. So in reality, the distribution does not range from $0$ to $1$, but from some number $k$ to $-k$, where $k<0$. The area under the curve is $1$.
A generalized distribution is
$$f(x, \mu, \sigma) = \frac{1}{\sigma \sqrt{2 \pi}}e^{-\frac{(x- \mu)^2}{2 \sigma ^2}}$$
$\sigma$ and $\mu$ can vary between particle type, though there is no continuous function connecting them. They exist in discrete types of your choosing.
The force from ikimgiir is probabilistic3, too, and it depends on a variable called $z$. You see, each particle has a different $z$ that it has its entire span of existence that is independent of $f$. The latter is a probabilistic distribution of the strength of the interaction whenever it happens; the former is a probabilistic distribution of when it will happen.
We describe $z$ using a cumulative distribution function, where $x \to t$ and the graph is renewed whenever a particle exerts the ikimgiir force. This function describes how likely it is that such an event has happened since the last time the particle exerted such a force (or since its creation), starting from $t=0$. Therefore, the probability at $t=0$ is, for a given particle, 50%.
So the ikimgiir force can be described in terms of $i$, $f$ and $z$ But we still need to describe its effects on other particles. When a particle exerts an ikimgiir force, it creates a spherical cavity. The size of this cavity is governed by a number, $a$, and $f$. $a$ is a universal constant, and represents the maximum size of the cavity (in the case of $f=1$). The radius of the cavity is $f \cdot a$. The final volume of the cavity is the volume of a 4-dimensional ball4.
The ikimgiir force acts on particles only when the cavity is expanding, and only on particles on the edge of the cavity. For particles at an instantaneous distance from the center of the cavity, $r_{inst}$, the force is
$$F_I=\frac{i}{r_{inst}^3}$$
where $i$ is the charge of the particle causing the cavitation. This is an inverse-cube law. Particles not on the cavity edge do not feel this force, but they are pushed away by particles touching the cavity.
Kaaziikkhaaku
This isn't a force, but a property. It describes how likely a particle is to combine with another particle (of the same time) and form a composite particle. $k$ is the same for all particles of a given type, and is just a probability (e.g. 50%, 78%, etc.). Kaaziikkhaaku only applies when two particle are within a certain distance of each other.
1 . . . which is what I have just done.
2 Names taken from a random syllable generator here.
3 Okay, so we've sacrificed a bit of determinism here.
4 I say "ball" and not "sphere" because in mathematics, "sphere" actually refers to the dimension of the outside boundary of what we would normally call a sphere. For example, the boundary of a baseball is a 2-sphere.
