That has a different meaning, I think. Saying "I used to be late to school" suggests that you no longer are late to school and now arrive promptly. Saying "I was late to school" generally implies that you're referring to a specific incident (i.e. "I was late to school yesterday").
As I understand it, if you are referring to a specific action that began and ended in the past (i.e. "I was late to school yesterday"), you would not use the Italian ímperfetto. "I used to be" or "I would be" are both acceptable ways to translate the Italian imperfetto into English, which does not have an imperfect tense. This is nicely explained in the opening tips section.
Yes, we Americans use both "to" and "for".
Both with different implications. "For" suggests one was late for an event (late for the show). "To" implies being late in regards to arriving at a location (late to the park).
For = school as an event To = school as a destination
English English speaker - I'd never say "I was late at school" to say I arrived there late. You could say " I arrived late at school" - subtly different! If you said "I was late at school" it is a slightly colloquial way of saying "I stayed till late at school"( e.g. because I had work to do there)