Sorry for necro, but from what I was reading there were a few points not addressed, it seems everyone was addressing the physics well, which is great, but we seem to forget how this would affect our little bundles of DNA that are gonna be making societies in these worlds.
1) I would say avoid huge massive stars that will burn out in a blink, but smaller and main sequence stars would work perfectly fine for our purposes.
2) Stars die when their CORE runns out of hydrogen and helium, and collapses. Beyond that I have no clue.
3) I dunno about it being that limited. I feel generating it at the start and putting "stars" in the sky that are other planets in the solar system would be more immersive.
4) They should definitely be a thing for 2 reasons: 1 they're just really cool and 2, that CAN happen, so why not? My bigger concern is whether they should be P-orbit planets (orbit the barycenter around both stars) or S-orbit planets (orbits one of the stars). Though to avoid bigger issue, habitable ones should be limited to smaller stars (red/white dwarfs)
5) Moons affect tides, which can affect aquatic civs. Having the homeworld as a moon has other issues, too. Bigger tides, basically being eclipsed for 1/4 of your orbit around your parent body. But it can be interesting, too. Pretty skies, lots of resources available from nearby planetoids and asteroid belts.
6) you got that handled pretty thoroughly
7) Size does play into mass, density, and available land and resources. We should put some variation here. Maybe pick a "mean" size, and make a standard deviation distribution so most habitable worlds are around that size, but not exactly that size.
8) same as size, it should vary, since that affects biology and air pressure, and so diversity. But not vary so much that it would kill a non-native from a similarly concentrated biosphere. So like O2 on Earth is 22%, it could range from 18~25%, same can be applied to other gases. Earth is our only baseline, we might as well use it, but we should't make it universally uniform, thats boring (cue exact same continents on 10,000 worlds)
9) I think a variable model by atmospheric density would work well, and allow a variety of atmospheric densities (and colors)
10) sounds good
11) We could set up a plot, I actually have an idea of "you" terraforming a barren world into a fertile one, to allow some degree of customization. You're limited in scope, you can select the star type, and weather you want moons, etc., but you can only customize the actual planet. Even then, that doesn't guarantee survival.
We could also make the detailed customization available only after one of your species reaches ascension, which i think was mentioned...