You'll clear immigration and customs for your carry-on in FRA, and customs for your checked luggage in FMO. So there is no need to collect baggage in FRA (nor would you be able to if you checked it through). The only thing you have to find is the LH transfer desk to get your boarding pass.

Pretty sure that CO arrives at terminal 2 - which means you'll need to take the people mover to terminal 1. You used to be able to do this pre-passport check, so that you could get over to T1 and go through immigration there.

Not sure if LH has a desk in T2 (don't think so).