Can a 'hyperspace' make sense if matter entering it is very quickly 'dumped' out of hyperspace at another point in our space.
That is to say, spaceship enters hyperspace and is almost immediately dumped at another point in space. This might be due to the hyperspace having a different set of spatial relationships relative to our own spacetime.
Don't use "dimension" to mean "pocket universe" - use it properly, to mean direction.
First, you move some distance ana (along the hyperspacial axis), then you move some distance forwards/backwards/left/right/up/down, and then you allow gravity to pull you back kata to the layer you started on.
This is a bit like using a plane or hot-air balloon to more up, then move forwards/backwards/left/right, and coming down again - or, alternatively, using a submarine to dive. (Some submarines, primarily models or hand-held propulsion devices, are always buoyant, and require constant power/thrust to remain underwater)
If the 3-dimensional layer that we perceive is curved when seen at higher dimensions, then the effect is somewhat like this:
Thus, the distance that you travel in our layer is magnified, even though you never exceeded the speed of light.
Of course, since time may also be curved, you might find, after moving ana for 30 minutes, then kata for 30 minutes to where you started, that only 5 minutes have passed, or maybe an entire year! Combined with needing to know which direction space is curved in at each location, this means that accurate "hyperspace maps" are required, and reliable transit routes are plotted out.
On the other hand, sometimes cutting through unusual space has benefits - produce dispatched a year in advance arrives having only aged a day, or doctors settle in for a 6-month cruise, but arrive mere hours after a planet signalled a medical emergency. Or, situate your prisons in a "slow" location, so that no one can stage a "hyperspace breakout" - and a 10-year jail term for the prisoner might be 50-years in 'realspace', allowing victims to move on with their lives in peace. Companies would buy "fast time" locations, to perform a year of R&D per month.
Aside from the "very quickly" part, what you have described is very similar to how jump drives work in the RPG Traveller. When a ship engages its jump drive, it is transported into an alternate universe/dimension/hyperspace, where it is isolated within a bubble of conventional reality. Approximately one week later, it drops back out into our universe at a location from one to six parsecs away from its original location.
As far as I'm aware, nobody has ever complained that this "doesn't make sense" (or at least it's no worse than any other proposed method of FTL travel), so I think you should be safe on that count.
There is a lot of leeway in creating hyperspace dimensions, since they are purely hypothetical. If the hyperspace dimension abhors matter enough to immediately spit it out, however, you would have to explain how matter can enter it in the first place (I would expect some resistance). There may be some minimum (and rather high) energy or pressure needed to push a spaceship into hyperspace, after which it is quickly squeezed out elsewhere. Sort of like pushing an air-filled ballon under water. You would also have explain why the spaceship doesn't just pop out where it was pushed in, or very close to it. It might also be difficult to decide where you pop out - some mechanism for guidance will be needed. Maybe the spaceship moves in the direction of the pressure applied to it to enter hyperspace, and this pressure must be maintained until the hyperspace 'decides' to pop it out.
Can a 'hyperspace' dimension
If we take it literally on the word "dimension", it could be your world has an additional semi-conventional space dimension. That is, everybody is embedded in three space dimensions that resemble our universe, but actually there is a fourth spatial dimension that can be acceded by machinery. So, no, you cannot look in that new direction... yet, it is possible to make machines that move in that new direction.
matter passing through from our world can't be present
That looks like a contradiction to me, if matter can't be present, then matter should not be able to enter. Let us tweak it by saying that matter passing through from our world can't stay (I think that is what you mean).
and is very quickly 'dumped' out of that dimension in another point in space.
Alright, I think we can solve this with topology. Let us say that the three main spatial dimensions are flat at boundary conditions (that is, they can still have curvature, but only locally). Yet, that is only when we see these dimensions at the default position on our new fourth spatial dimension. When we consider the 3D slice of conventional space at some other position on that new dimension we find global curvature such that a straight path (think: geodesic) will return objects to the default position.
That also explains why there is a default position for this extra fourth spatial dimension: it is a stable equilibrium, small perturbations bring objects back right away. Note: this implies that there is nothing special about going on one direction or the other in the extra dimension.
Addendum: If it weren't symmetric on the new dimension, this model would suggest that everything is perpetually moving in a certain direction in that additional fourth spatial dimension making it work more like time in that it goes inexorably forward, then we are talking time travel... well, I suppose that FTL travel is time travel anyway... hmm...
This also has the consequence that you need to consume energy to stay out there, otherwise you are thrown back to conventional space. Now, how far it will throw you depends how you move, both on conventional space and in the extra dimension. So you could travel the same distance by going very fast and no use of the extra dimension, going not so fast and doing a shallow leap in the extra dimension, or going slower but diving deep in the extra dimension.
perhaps due to this other dimensions 'different shape' relative to our own
Ah, you well onto it already.
However, I am not really sure on what the shape is. On one hand, the more condensed space is, the more likely things will go there, suggesting that space out there is “wider”, on the other hand if more direction lead to conventional space, that suggest that space out there is “narrower”.
First off, this is not possible. there is no "hyperspace" dimension. Sounds a lot like Star Wars baloney.
There are four dimensions. We live in the third dimension, where there is length, width, and height. Hence the name three dimensions. Though two dimensions can be displayed in a three dimensional universe, a three dimensional being cannot enter a two dimensional universe (breathing and digestive tubes rely on three dimensions to function, so if you went into a two dimensional universe, you would suffocate.) Plus, even if you found some universe-shattering loophole, there would be no way to travel faster there than here. In short, there would be no point in doing it.
If you want to get similar results, however, try converting your ship into neutrinos, these can pass through normal matter like there are nothing, though collisions do rarely happen. How to encode your and your ship's data into neutrino patterns could be simple, though you would have to have an extremely dense receiving point to later encode the data and rebuild you and your spacecraft.
