Yes, CO and LH have an interline agreement. So, your bags will be checked all the way through to your final destination. There's no need to pick up your luggage in FRA. Just make sure to show the check-in CO agents your onward LH ticket. The LH ticket can be a completely separate ticket from your CO ticket, and CO will check the bags through anyway.

When you land in FRA, you do not need to go through the regular Arrivals/Immigration lines. Just follow the signs to your next destination. You will pass through a transfer immigration line (which normally is a short line) to go to your onward LH flight. This immigration line is just a stopping point as this is your first EU stop - and that you are continuing to another EU stop.

And, yes, you will have to stop at a LH transfer desk to pick up your LH boarding pass.