If you misconnect in FRA on two seperate PNRs LH will not take responsibility. You will have to rebook subject to the fare rules. In these cases it is always best to book both tickets into one PNR. Obviously this is only possible via a Travel Agent. Then the airlines involved cannot claim that the flights were unknown to them.
I am not sure if your travel insurance would cover a misconnection ? I doubt it, but check the small print anyway ?
I have no idea if LY has an interline agreement with LH for baggage?
Happy travel through FRA !