Alternate Planets, Suns, Stars, and Solar Systems Thread

How far can one twin orbit the other without putting either one at risk of tidal locking?
You can calculate it provided you know the parameters. Finding out where they will tidally lock lets you kick up the semi-major axis to beyond that, which nets you various orbital resonances. An orbital distance above the tidal lock distance but within the Hill Sphere is what you’re after, right?
Would both still have the axial tilts needed for seasons, or would the gravity from either side straighten them up?
Solved easily by simply giving the barycenter of the system an axial tilt. Then you can even have them have no tilt relative to one another.
How will orbiting each other affect the seasons themselves?
The Moon orbits the Earth and seasons are still seasons. Having the barycenter that far outside the Earth is going to more greatly affect the weeks, rather than the seasons. And even then, not by much. The Earth-Moon system diameter is smaller than the variation in the annual Earth-Sun distance, and though this would be larger, I can’t see it being significantly larger.
Will a year still last months as our Earth does, or will it last weeks, as is the case with moons?
A year will be a year, because it’s a year. Axial tilt defines that.
Will a binary planet render the use of a moon moot?
I’m not sure how you’d get one stably around either planet. Assuming no tidal locking as you said, you’d still have tides on each planet like a regular moon would provide… provided that the shared resonance is high enough for that. You could have slower tides, I suppose.
How bright will a binary planet be on the night sky?
Easily calculable as a direct function of finding out the first calculation above. We know the absolute magnitude of Earth, so all you have to do is plug in the distance you need from above and calculate the relative magnitude at that distance.

Workable can correct me, of course.
If Earth were just the one twin of a binary planet system, then the following questions are as such:
Without giving away too much, I've been working on a binary planet scenario for the MotF, so I might be able to give some insight here that hasn't already been mentioned.
How will orbiting each other affect the seasons themselves?
This explanation is for a tidally locked pair. The actual seasons are going to be similar, as mentioned, but there are interesting phenomena that arise during each of them. If they're not tidally locked, it's probably not too different from Earth.
Will a binary planet render the use of a moon moot?
If the planets are tidally locked and relatively close, then you could probably fit a moon stably within their hill sphere. This would be helpful, because with tidal locking, you don't get a monthly/daily tide cycle. As already mentioned, if they're not tidally locked, you probably can't get a moon.
  1. How bright will a binary planet be on the night sky?
Earth is 2.5 times as reflective as the moon (moon = 0.12 albedo, earth = 0.30). 2.5 * r^2/a should give you a good answer. r = planetary radius in lunar radii, a = distance between planets in lunar distances. This tells you the brightness of a full *planet, compared to a full moon.