As I see it:
1) Yes, you will go through an immmigration check in MUC as you pass from gate to gate. Very simple and fast.
2) Bags go to CDG, Customs is in CDG (done by using different luggage tags and a common baggage hall. Schengen bags get a tag with a green border stripe).
3) Bags checked through.
I think you will be pleased with how smoothly things work at MUC in the new terminal.