I think you are assuming that the planet-moon orbital plane lies within the planet-sun orbital plane.
I am assuming the usual mechanics occur, which will tend to drive the moon onto that plane, even if it doesn't start there.
Either way, though, "short cosmological" time is plenty for an intelligent species to evolve. Our own moon will fly away in "short cosmological" time, but that doesn't mean a heck of a lot to us brief candles.
"Short cosmological" is more on the hundred million year range or less than the billion year range. An object like Pluto (small, very far from the star) might take 10 billion years or more to come to be tidally locked. An object like Mercury (small, close to the star) has reached it's steady spin-orbit resonance (locking is just a very simple such resonance) in well under 4 billion years. An object like our moon (large on the scale of the two bodies in question) got tidally locked early on as well.
Can you place a game in a time during the process of locking? Sure. That can be cool because it would be... geologically interesting (earthquakes, volcanoes, tidal waves, and so on). But the math would suggest that the locking would occur early in the arrangement, rather than later.
Regarding weather: it's quite possible that you'll see local bands of convection rather than just a planet-long low-altitude "sunward" wind under a planet-long high-altitude "nightward" wind.
Yes. That "low/high" pattern is only the general flow that's required. You can get to that with convection bands.
How big can this fraction get? Could it be something as big as 37 moon rotations for every one of the planet's rotation, making the moon rise every 24 earth hours?
In general, the system will be driven to the closest state of strong resonance. But the relative sizes, speeds, compositions and distances between the object matter a great deal.
I'm under the assumption that the same tidal effects that straighten the planet's axis will also pull the moon to orbit at the planet's equator. This leads me to believe a moon would net constant eclipses.
The moon would transit the sun frequently. Whether that counts as an "eclipse" depends on relative sized and distances. Our setup is peculiar, as our moon's just at the size and distance to just barely cover the sun's disk in the sky. But that is a coincidence, not a thing that generally happens.
I'd still like to know if anyone knows how big the star would appear in the sky.
I'll see if I can find the data to tell you. I suspect it'd be much larger than our sun is in our sky.
Edit: Quick back-of-envelope calculation, given what data I can find on Gliese 581 and 581g : Our sun has an apparent diameter of about 0.5 degrees when seen from Earth. I think Gliese 581 would have an apparent diameter of about one degree, as seen from G. So, something like 4 times the total area - big, but not super-jigundo.
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