Now is everyone
absolutely sure a tiny creature gets to REMAIN in the occupied square after its turn is over?

I thought this was the case, but after
this thread about the same case in the similarly worded Pathfinder ruleset, I'm not so sure.
Here is an older thread on the matter too.
Yeah, Frank, you're right. I meant to mention this earlier but forgot.
Per PH, p. 128:
"A Fine, Diminutive, or Tiny creature can move into or through an occupied square. The creature provokes
attacks of opportunity when doing so."
Yet technically, on the same page (above it):
"You can’t end your movement in the same square as another creature unless it is
helpless"
However, as a DM I would ignore the second part in this case, or rather choose to think "move into" implies "can end" as it seems more correct. Likewise, there is, I believe, no technical limitation on the weapon used on your space. However, I wouldn't allow a PC to use a two-handed sword on a rat in his space. Likewise, if the rat were actually on the PC at the time (dropped from the ceiling, etc.) the PC would have to take a grapple action to get it off (knock it off, grab it and throw, use a dagger, etc.).
As to the AoO. AFAIK, there is nothing in the rules that says a 0' threat range means no AoO. It simply means that you can't threat OUTSIDE of your space. So, you may AoO inside of your space.
Think of it this way: take four adjacent spaces, the PC in one of them threatens the other spaces, correct? He doesn't threaten (normally) the space beyond the adjacent space.
Now just shift everything down. A Tiny would threaten, in a Tiny manner, everything adjacent to his Tiny space (e.g. a regular space) but not beyond it. (ignore the idea that the Tiny creature would by default be adjacent to three other, non-Tiny spaces!)