You're going to need to act a little selfishly if you hope to get through this with a favorable result... Here's my take.
Your mom should absolutely yell at your dad on your behalf - without you needing to ask. If your dad doesn't put his wife in her place (her place being: not your mother, and not an authority figure over you) then it shows a clear disrespect for the fact that you're his daughter. You, as his daughter, should always 100% come before other people, and his wife is "other people". If he's not willing to acknowledge this, then he clearly has no interest in being your father.
The core problem here is as follows, as quoted from your original post.
his wife told him that he could never see me or my mom again.
His wife doesn't have a say in the matter. Tell her to fuck off. He's your father, and she's a stranger - she doesn't get to dictate that stuff. The fact that she tries to appeal to your decency when she has none of her own is disgusting.
Answers to your questions (I am not a therapist, nor have I personally been in a similar situation to yours, but here's my two cents as someone who doesn't take shit like this letter from people):
1: Stop feeling like a stupid piece of pond scum. You're a teenage human female with teenage human female emotions. The only people who should be feeling bad here are your father and his wife. Talk to your therapist. Learn to be blunt and honest. Stand up for yourself.
2: Yes. Your mother should absolutely be in contact with your dad, about this and about other things. If yelling happens, so be it.
3: Don't send a thank you card. That's passive-aggressive and won't accomplish anything. Instead, call your dad and ask if he knew about the letter she sent. I'm willing to guess (in his favor) that she sent it without consulting him and that he wasn't aware of the contents. If you can't call him, write him a letter, enclose a copy of what his wife sent you, and require a signature with ID on delivery (even though that sounds extreme). You need to know where your dad actually stands on this issue.