Yes, the cosmological constant is present around black holes. The amount of the pressure isn't enough, though, to stop the formation of the black hole.
Yes, gravity does work at the Big Bang. In fact, it is the equations governing gravity that leads to our understanding of the BB.
Sorry, but this doesn't work. You can't get a consistent, black-body radiation to within 1 part in 100,000 from stars in this way. There are simply too many different temperatures for the different stars, which would lead to a spread in the detected radiation that is contrary to the actual observations.
The matter-antimatter asymmetry happened before the time of nucleogenesis. Since we are still investigating the specifics of CP violation (leading to that matter-antimatter asymmetry), but we know it exists, it is more a matter of the *amount* of matter left matching the observations than anything else.
This is an area of active research in the particle physics community. But the basics is that the symmetry between matter and antimatter isn't perfect. We know this from the differences between the K meson and the anti-K meson. And, just recently, we found CP violations in reactions involving neutrinos (which is far more likely to give the correct amount of remaining matter: 1 part in a billion).
https://phys.org/news/2020-04-matter-antimatter-asymmetry-t2k-results-restrict.html
Not an idea that remotely fits with the observations. Stars alone cannot produce the specifics of the radiation that we observe.