Anyone have experience connecting internationally thru Nagoya albeit with checked luggage and 2 different airlines? Would I have to go thru customs after I pick up my bags? How long will it take? I'm looking at less than 3 hours from landing time to departure time.

You might want to call one of the two airlines you will be travelling on to see if they have baggage transfer agreement. If they do, then you can check your bag at your point of departure all the way to your final destination without having to reclaim and recheck your baggage at ICN.

NOTE - I have not done a connecting flight here, but I have used NGO. I think that 2 to 3 hours will be enough time to make any kind of connection at NGO. Everything is under one roof.

(It is do-able at Narita which is a much bigger airport with two terminals).

The airport website offers some information on connecting internationally (i.e. from one International to another International flight):
http://www.centrair.jp/en/guide/int-int/index.html

From the airport website, it is apparent that you don't need to go through customs or immigration. You do have to go through a security check point. This is consistent with other international airports in Japan.

As per the previous poster's advice, you might want to call your airline for additional assurance that the checked luggage will be transfered to your next flight without you needing to do anything. (This is standard practice at other international airports in Japan).