A.V.
Here's the way I think about it - there's his story, her story and the truth in the middle. What goes on between them should mostly be between them.
When my mother complained over and over about her now XH, I said, "You know what I think. I will support you but I can't be your sounding board and I can't fix your marriage. I suggest you find a counselor or a lawyer or both." And that was harsh, maybe, but after a few years it's just not helpful to anyone to hash and hash and not change anything.
You need to figure out how to have healthy boundaries with her because she's dragging you through her unhappiness and ruining what positives you might have with your father. And have you ever considered that your dad is unhappy with your mom, but just not telling you about it? See the start of my post. I agree that they need to be talking to each other and not to you or to anyone else outside their marriage other than a counselor. And maybe that's what you need to say to them. If she is depressed, she needs professional help. That is about HER and not YOU.
Please do NOT send a copy of anything to this other woman. It's not your place. You may also be reading into it more than is there. My friend is writing a book. She's befriended one of her sources. She adores her DH but she is fascinated by this source guy and his story and they get along well. But at the end of the day, they laugh and send a few emails and go back to their spouses. And if your dad is having an affair, there is likely little you could say to her that would be productive or really fix the root of the problem.
Bottom line: Don't let your mom poison your life if she's unhappy in her own.
ETA: Others have good points about your parents being different people and your mom's effect on the family. My MIL is a worrier and has become a gatekeeper as they age and deal with health concerns. She will say that FIL isn't up for company, but if we swing by to drop something off, it takes an hour because he wants (needs) to tell stories, talk to someone who isn't Debby Downer, etc. Your dad may indeed be a grumpy old man. But he may also be a man from an era where men made the money and women kept the house and he wasn't expected to be warm and fuzzy. The joke is that my FIL's version of "I love you" is "How's your car running?" Do you ever talk to him about his writing? Maybe that would be a good thing to try, because it's obviously important to him.