Couldn’t they readily identify a 365-day cycle just from the background stars, i.e. when Pisces (or whatever they call it) is overhead (or nearly so) every twelfth “midnight”.
How 'bout “Mooners?” 
13th midnight, but yeah. They definitely could. At the most rudimentary level, you could see what stars were on the horizon at the time of your 28-day “day” when nothing has a shadow. That would change throughout the year.
Sure there is . . . but it keeps changing, in 28-day cycles.
I think it’s because there’s no gravity on the moon.
Not now, but there was until 1959.
And don’t we have to add on another 1.5 degrees or so of axial tilt relative to its own orbit?