Yes, you did eat. But since you really were hard pressed for time, you had to eat really quickly ! In some contexts however, this can also mean that you did not eat. e.g. "After I was done with my homework, I hardly had time for lunch. So I had to run to the classroom grabbing a sandwich (which too, I did not manage to eat)"

Depends on the situation and the intonation. She could be saying why she didn't have lunch in a sarcastic way. Or she could be saying she had to hurry her meal.

You ate. But you had to swallow fast!! On the other hand this can be a thought that goes through someone's head as he contemplates whether to bother with lunch because he knows it will be such a hurry. It depends on the context.