How do I cope with my dad going to a nursing home?

Wow. Your father is a hypocrite and a bitter old man so try not to feel too bad about all this. Nursing homes are an inevitable part of life for many people and its not your fault that he has to go to one. Give yourself some time and try not to judge yourself or your mother for her actions. Hopefully in time he will accept his fate and you can go visit him, but for now I say let him be.