Frame Challenge
Rule 1 has got to go.
The brute force strategy is required to find aliens, WCPs, or anything else. Just keep opening as many WCPs as you can to find whatever you're looking for. The problem is that everyone is going to be doing this. No matter how you distribute WCPs, avoiding paradoxes by arbitrarily preventing light-cone intersection has two big problems:
Unreachable Aliens
Ordinarily, the assumption in sci-fi is that it is much easier to find big advanced civilizations than small ones, because they are much bigger and louder. Humanity is entirely invisible from more than a hundred light-years away, but a million-year-old Dyson swarm could be spotted at a distance of, well, a million light-years. Since space is 3-dimensional, the volume of space from which something is visible scales cubically, so that Dyson swarm is visible from 1 trillion times as many planets.
Rule 1 totally flips this on its head by making space exploration mutually exclusive. If aliens open a wormhole within our Hubble horizon before we open one within theirs, it will be the only wormhole that can be created between us, and we won't know where it is.
This means it is actually much, much harder to find more advanced civilizations than less advanced ones—WCPs provide an easy way to find untapped resources, so any advanced civilization will have been opening many, many WCPs, constantly, for a very, very long time. The odds of us making the connection first are slim.
If a WCP is opened between the Hubble horizons of any two civilizations A and B, the probability $p_{A}$ that it was opened by Civilization A equals the ratio between the number of WCPs opened by Civilization A and the total number of WCPs opened by both.
$p_{A} = \frac{N_{A}}{N_{A}+N_{B}}$
The number of WCPs a civilization has opened can be expressed as the integral of the rate at which they open WCPs over the time period since they started opening them.
$N(t) = {\huge\int}_{t_{start}}^{t_{now}} \! R(t) \, \mathrm{d}t$
In other words, civilizations even slightly older than us will have created such a larger number of wormholes that it will be almost guaranteed that the single wormhole created between our horizon and theirs would be created by them. The universe is very old, so any civilization which already exists is likely to be many millions of years older than us and will be vacuuming up WCPs at an unfathomable rate.
Also, keep in mind that the Hubble horizon distance is 13,000,000,000 light-years. The vast, vast majority of wormholes connecting the horizons of two civilizations will not actually result in them finding each other, since there are no other methods of superluminal travel.
An Example
Let's do some napkin math. For simplicity, we'll assume the following:
- All civilizations increase their power generation by 1% per year
- All civilization discover WCPs at around Kardashev 0.8 (for humanity, this is around 2200 CE)
- The proportion of all civilizations' power spent on wormholes is roughly the same
- The energy spent per wormhole is roughly the same
With these assumptions, $R(T) = k \cdot 1.01^T$, where $T$ is the number of years since discovering WCPs. This gives us formulas directly relating the age of a civilization with the number of wormholes they've created.
$N(T) = \frac{k}{\ln 1.01} (1.01^T - 1)$
$p_{A} = \frac{N(T_{A})}{N(T_{A})+N(T_{B})} = \frac{1.01^{T_{A}} - 1}{1.01^{T_{A}} + 1.01^{T_{B}} - 2}$
Let's say Civilization A is humanity, and Civilization B is another young civilization which discovered WCPs a millennium before humans (in 1200 CE). By the start of the 24th century, one hundred years after humanity's first wormhole, a wormhole created between the Hubble horizons of Earth and Civilization B has a 1-in-300,000 chance of having been created from the human side. These will gradually improve, but they will never exceed 1-in-21,000, because exponential growth means that Civilization B will always be around 0.4 Kardashev ahead of us. Normally, advanced civilizations' growth rate would slow down over time due to the light speed limit, but with WCPs that's not an issue.
A civilization being such a similar age to our own is extremely unlikely. Every thousand years of separation reduces $p_{A}$ by another factor of 21,000. If Civilization B is a million years older than us (which is still very close), the odds are roughly $10^{-4,300}\%$.
We are not finding any aliens.
Causal Separation Collapse
Let's pick some arbitrary numbers. The largest nuclear explosion in human history, Tsar Bomba, released 200 petajoules of energy. A modern nuclear power plant would have to run for seven years to produce that much energy.
Assume that it takes that much energy to open just one wormhole large enough to see through. In the setting of your story, let's say humanity uses about 10x as much energy as it does today, and that 0.1% of that is used for opening WCPs.
In this scenario, humanity would open about 20 wormholes a year. We would almost certainly find nothing of value on the other side of any of these wormholes. Nevertheless, with each wormhole opened we expand the volume of our observable universe by an additional Hubble volume, which is $361 \, \text{Gpc}^3$, or $12.5 \times 10^{31} \, \text{ly}^3$.
When a civilization starts opening WCPs, they are essentially building a connected graph of wormholes. If I create two wormholes, both of which connect our solar system to locations previously outside of Earth's future light cone, those two locations are now also casually connected, through Earth, like nodes on a graph. The more wormholes we create, the larger our graph becomes, but it will always be a connected graph because one end of the wormhole will always be within our observable universe.
Let's consider that civilization from earlier which was a million years old and opened around $10^{4,000}$ times as many WPCs. That means that their observable universe expands by roughly $10^{4,000} \, \text{ly}^3/\text{yr}$. To understand how big of a number that is, if we change the units from "cubic light-years per year" to "Hubble volumes per femtosecond", the exponent is still about 4,000. Yet, give it another million years and it becomes 8,000. Exponential growth, am I right?
Keep in mind, this is a young civilization on cosmic time scales. A billion year old civilization will be eating around $10^{4,000,000} \, \text{ly}^3/\text{yr}$. That's a lot of space. So, what's the issue?
Rule 1 says that you can't create a second wormhole to a region that is already within your observable universe.
Ancient civilizations will swallow the entire universe into one single causal bubble and WCPs will become completely inert. This will almost certainly happen before life on Earth ever crawls out of the ocean. Here is why:
If Civilization A creates a wormhole to a location inside of any of Civilization B's 13,000,000,000 light-year wide observable bubbles, their two previously-disjoint universes are now one casually-connected observable universe within which neither can ever create another wormhole.
When civilizations start popping up and making wormholes, they will eat up huge volumes of the universe. If the universe is infinite, there is an infinite amount of volume to eat up. However, there are also an infinite number of civilizations eating it, and this infinity is growing.
Once the oldest civilizations reach a causal volume larger than the civilizations-to-volume ratio of the universe, it's over. We don't know how early life could have started forming in the universe, but we know it's at least 4 billion years ago. By the time humans show up, the oldest civilizations' causal volumes will be at least $10^{10,000,000,000,000,000,000,000,000} \, \text{ly}^3$, each. Unless the universe is so sparsely populated that civilizations are, on average, at least that many light-years apart, we will be too late to make wormholes at all.
Solution
You need a different way to prevent paradoxes. Fortunately, wormholes are only actually "fertile ground" for one paradox, time travel. Subjecting one end of the wormhole to time dilation (e.g. via a gravity well or relativistic speeds), the two ends become temporally de-synchronous and travel through them becomes time travel.
All you really need to do to fix the causality problem is have wormholes operate in an absolute frame of reference, temporally at least. I know, Einstein is rolling in his grave, but it's sci-fi not sci-reality and frankly this requires less suspension of disbelief than Rule 1 anyways. If you just say that wormholes are always in sync, there's no way to time travel and no paradoxes can happen.
Without Rule 1, both of the big issues go away. Ancient civilizations will still dominate the entire universe thanks to the lack of a speed limit, but they might not notice you so that's fine.
A Side Note
You stated that you initially rejected the "WCPs are dark matter" idea.
Originally, I thought that WCP could resemble dark matter, but it seems like its distribution (as dark matter is thought to be) would leave no chance of opening a portal into anywhere except galaxy centers and other places that are not friendly to any kind of life.
This is false. As we currently understand it, dark matter is primary concentrated in halos around galaxies and galaxy clusters. Or, more accurately, galaxies are primarily concentrated in clusters of dark matter--dark matter is the driving gravitational force in the universe, and baryonic matter is sucked into its wells.
The baryonic matter of a galaxy typically occupies a much smaller volume than its dark matter halo. Furthermore, dark matter is distributed across the intergalactic space within a galaxy cluster, and along the filaments between clusters, so a randomly selected dark matter particle is almost guaranteed to be in the absolute middle of nowhere, with maybe one-in-a-thousand being inside of a galaxy. Queue the numbers game.
