I apologize if this seems a stupid question, but neither astronomy nor geometry are my strong suits.
If I stood at a fixed point the moon’s equator and observed the sun, how long would it take to complete a journey across the sky (a “day” so to speak)? What would my observations of the Earth be like - would they be a mirror image of what an observer on the Earth’s equator would observe in the moon (waxing and waning as the month progresses)?
The Sun would move across the sky in about 28 days - but the Earth would remain nearly fixed in the sky, moving only very slightly (see Libration - Wikipedia) - but you’d see the Earth rotating every 24 hours, so over the period of a day, you’d sometimes see America, sometimes Europe, and sometimes Asia below you. The portion of the Earth that is lit by the Sun would mirror the phases of the moon - when the Moon is seen as full from Earth the Earth is seen as “new” on the Moon (and vice versa).
The Earth is four times larger in diameter than the Moon, so yes. For reference, the Moon’s area viewed from Earth is about 1/3 the area of Africa as viewed from the Moon.
ETA: Just to be clear, by Moon’s area, I mean its cross-sectional area.
Actually you’d see one rotation of the Earth every 24h 50m. The Earth has to rotate a bit more to catch up to the Moon’s motion.
The Sun would take 29.5 days to go from sunrise to sunrise. Half that to cross from horizon to horizon. Fairly slow motion but you’d notice it over the course of a day, err, um, I mean 24 hour period.
As mentioned above, except for minor wobbling, the Earth, if visible, will remain in the same spot indefinitely. You’ll see it spinning, but it will always be at the same position in the lunar “sky”. If it starts moving across the sky, then
Also note, by the way, that the Earth as seen in the Moon’s sky would be about four times bigger across than the Moon as seen in the Earth’s sky, and much brighter.