I'm trying to think of a circumstance when I wouldn't root for a Big 10 team in an OOC game and I just can't come up with one. Let us not forget that with few exceptions these are all away games for the Big 10 and that they are usually paired against someone ranked higher.
As for rooting for or against Michigan, my take is that like it or not, we are linked to them historically. You have to be older than me... and I'm a geezer... to remember when the Big Game was Illinois. I want to beat them every year and the reason for that is because, like OSU, they represent one of the best programs in the NATION year in and year out. We measure our program by how we do against them. That's why we don't get worked up for the Indiana/Northwestern game.
Let Michigan get blown out in a couple of bowl games, loose five or six straight to ND, loose four straight to MSU and see what happens to OSU's rankings and to the rep of the Big 10. I respect Michigan academically and athletically. I support them in all OCC games and I cheer like hell against them everytime they play Ohio State. Go BUCKS!