In my opinion, a month is an awfully long time for a two year old. I would try first for a few days and work up to it.
NO way. A two year old is still a BABY. They are not old enough to completely understand it is a just "a visit" .. I would NOT allow a child that young to to go MIL's. No Tell them- when she is older.
Nope. Maybe a week. She could file for abandonment unless you have it in writing that's only a vacation, too. Not going to happen and especially not at that age. It actually can make you look bad -even if you trust her. You just never know.
I wouldn't let my 2-year old be away from me for a month.