this is of course all in hyperbolic geometry...
These last two are me just more checking to see if my thought process is correct. thank you for any help!
3. how to prove that two parallel lines are not everywhere equidistant. this seems pretty simple to me given the angle of parallelism. you can just maybe create a triangle from the angle of parallelism and show that the distance is not always the same?
4. prove that two parallel lines have at most one common perpendicular. i think i know how to prove this one but I thought I would check. just again with the angle of parallelism, we can suppose the opposite, and create two perpendicular lines, and by one of the propositions where the opposite angle must be bigger, the next perpendicular line must be acute.