Am I to blame for their problems?

My first boyfriend recently got in touch with me after five years to say that he wanted to meet for a cup of coffee. I agreed, and he told me he was engaged to be married and wanted me to meet his fiancé. When I asked him why, he said they had broken up a few times because she thought he was still chatting with me, and he wanted me to resolve any misunderstandings. I agreed, even though I thought it was an odd request. When his fiancé called, she began to accuse me of trying to break up their relationship. I told her repeatedly that I have barely kept in touch with her boyfriend since we broke up five years ago. All we have done is like or comment on each other’s Facebook posts. She said she didn’t trust him and wanted to end their engagement. I hung up and am now trying to understand if I could have done anything to prevent this because it feels as if I am to blame in some way. What should I do?
I’m not sure I understand why this is your problem. You haven’t done anything wrong and have no reason to feel guilt or take on responsibility for how they are dealing with their relationship. This is between the two of them and you have put across your side of things very clearly. Speak to your ex-boyfriend, ask him to deal with this on his own terms, and disengage.

I want to ask a girl in my building if she will come to a movie with me, but her father doesn’t get along with my parents which may be a problem. They got into an argument a few years ago, and things have been unfriendly between our families since. Should I just give up on the idea?
That depends upon how this person reacts, and whether or not she has any interest in accompanying you to the movie in the first place. Why not ask her out first?