That is solely your theory, and completely unsupported. Basing an estimation on what a character could have done based on your own belief about how the Myrdraal interact with reality is highly suspect. And seconding that, four myrdraal can allow him to only appear vaguely in a flickering manner. By that logic, a single myrdraal could not give him the grounding to manifest even more strongly. The simplest explanation for why Ba'alzamon did not come to Baerlon is that he could not.

Just because he can find Rand's approximate location does not necessarily pinpoint him to a particular building.

No, it worked crappy, like your reading and logic skills. They were able to figure out what city and what section of the city the Bowl was in, but they lacked sufficient data to just swoop in and pick it up. As YOU point out, finding a stationary object should be much harder than locating a moving person. So if a dream search had such incomplete results for the stationary object, how on earth can you possibly expect it to allow a bound individual who shows no ability to truly manifest in the physical world, to track down a moving person, when we have already seen that it isn't even that simple for a stationary object? That IS the question I was answering after all - I was explaining why Ba'alzamon didn't just pop into the inn at Baerlon and grab the ta'veren. All you are doing is providing backwards logic that in no way refutes my contentions.

The rats have not been seen. Absence of evidence is not evidence of absence. Just because no one saw the rats (at night, no less) does not mean they were not there. In such pre-industrial times, with no heremetically sealed containers or expert use of poisons and cheap mechanical traps, people expected a certain amount of rats and other vermin. They did their best to prevent it, but when you store food in burlap or wooden containers, they ARE going to find it. The cats and other measures simply attempts keep the problem down to a manageable level. Even in the big city, Master Gill claims that Moiraine's ward have given him the only rat-free inn in Caemlyn, but his is not even one of the nicer or more upscale establishments. The impression given is comfortable to middle class. Plainly rats are a universal problem, and buildings with no rats are the exceptions, and those only a result of Aes Sedai intervention. So yes. The Stag and Lion DID have rats. Where the hell do you think all the dead ones come from? The problem the cook and innkeeper had was that their corpses were turning up all over the place. THAT is the unsanitary condition - not that rats were present, but that dead rats are being left about. The cook was not defending her cat's diligence in keeping them out of the inn, but rather its proper behavior in disposing of them.

You don't pay attention even a little bit when you read, do you? That's not how it works for wild animals. They have reflections in tel'aran'rhiod, and it is not only when they sleep. Do you think domestic animals stop dreaming? Or that physical objects fall asleep if you leave them in one place for a long time, which is why they have reflections? Why would lions be in their cages in Tel'Aran'Rhiod at night if they only went there in their sleep? The rules are different for animals than they are for people.

But the rats would have been able to resist the same thing happening to them?

That is too ridiculous for words. Why scatter them around the inn, and not the beds of his targets?