Personally, I'd say 'I'm sorry for being late' but 'I'm sorry I'm late' would also be OK. We tend to say 'I'm sorry for doing something' when we are apologising. e.g. I'm sorry for crashing your car. We tend to say 'I'm sorry to do something' when we are expressing sympathy. e.g. I'm sorry to hear your dog died.

The sentence is structured correctly, but most native English speakers would say "I'm sorry I'm late".

It's more common to say "I'm sorry I'm late". "I'm sorry to be late" is grammatically correct but sounds a bit clunky and old-fashioned.

Both are correct. The first is matter of fact and theoretical. In it you state that you are sorry about your lateness. It sounds rather passive. In the second, you take direct responsibility for your own failure. By making yourself the subject ("I am late") you are making no excuses whatsoever.