Does the "When Kalonian Twingrove enters the battlefield..." trigger for me and thus I get 2 of them?
If 1. is true, do I only have to exile the first Kalonian Twingrove token by Mimic Vat on the next end step and get to keep the cloned token from it's "...enters the battlefield..." trigger?
The rulings on Mimic Vat say:
Any enters-the-battlefield abilities of the exiled card will trigger when the token is put onto the battlefield. Any "as [this permanent] enters the battlefield" or "[this permanent] enters the battlefield with" abilities of the exiled card will also work.
This means that you will get one token copy of a Kalonian Twingrove, and one Treefolk Warrior token. The Treefolk Warrior token won't be exiled, as it wasn't created by the Mimic Vat.
This is assuming you have any forests in play. If you do not, then both tokens will immediately die for having 0 toughness and vanish!
Yes. It entered the battlefield, so its ability triggered. It doesn't matter how it entered the battlefield. It doesn't matter that it didn't exist before appearing on the battlefield. It wasn't on the battlefield, and now it is, so its ability triggered.
Correct. Nothing instructs you to exile the token created by Kalonian Twingrove, so you don't.