It's hard to say for sure, but it seems like the woman discussed went along with something that she didn't want to at first. Another way to say it might be "she gave in." Again, I'm not 100% sure of this because I don't know who "she" is, and I don't know much about the situation from the context your provided.

I think here it means, "she gave in". This means that she gave up an argument and took the course of action that the person she disagreed with advocated or desired.

You should verify this with some other English speakers, but I think the term "folded" can also be used when a business of company goes out of business, fails, or is liquidated.

In card games when one "folds" he realizes he cannot win and gives up playing thereby letting the other person win without continuing the game. In the context of your question she realizes she cannot win the argument and is reluctantly letting the first person have what he wants. It means she doesn't agree with him but she is letting him win the argument anyway.

Or, "she capitulated". "She surrendered" means somewhat the same thing.