ICN is really nice and easy and quick for international connections.
Never tried the LH/Ca combination for checking luggage thru, but LH has been good at this to me in the past - even on seperate reservations and seperate tickets.
Just make sure that you have printouts of the reservation showing the reservation number and also the ticket number and have a printout of the ticket information (if not a paper ticket).
If luggage is not checked thru you need to go thru immigration, pick up the backs, check in, back thru immigration and security. Also not a big deal, because it is typically quick and efficient.