Do, you also need to understand the idiomatic meanings of happen followed by an infinitive.
In #1, the expression "happened to" implies a coincidence:
- She happened to walk in the door jus as I was getting ready to go see her.
- I happen to be reading the same book that you were reading.
At other times, it can be used to challenge a false assumption:
- He claims he was at work yesterday, but I happen to know he was at the race track.
- I don't care whether you think it looks good on me. I happen to like wearing a fez!