Geez. I'm really sorry to hear all of that.
You don't have to take his name off of anything. Just try not to blame yourself for it. It's easy to go back and think about how you could've done things different, but isn't your fault.
My mom went through breast cancer a few years ago, it scared everyone and I didn't know how to handle it. She just wanted love from her family, though. If your mom is trying to get you to forgive her, it isn't hard. Maybe she doesn't deserve it, but she'll need that, and your siblings will need someone to lean on.