You'll need to enter Korea to transfer between ICN and GMP, you won't need to enter Japan to connect to FRA. There shouldn't be a problem with checking in at GMP for FRA-USA whether you have an GMP-HND boarding pass or not. I've had overnights before at ICN (noonish arrival and departure the next morning) and my bags were checked through and I got my BP for the departure leg from the lounge since my I couldn't check in at the hotel until early afternoon. There may be a minor BP issue on the GMP-HND leg since you'll need to enter Korea. All of the major airlines in both Korea and Japan have online check in and BPs so you can check in when you get to Korea.
No experience with an I-I ICN to GMP connection but I've always had to check in at GMP for the domestic leg.