Question

1. He was late for school.
2. He was late for the school.
(What is the difference between them? Does #1 mean that he was late for his school? Does #2 mean that he was later for another school, not his school?)

3. He was lare for class.
4. He was late for the class.
(What is the difference between them here?)

Answer 1

The reference "the school," as I wrote at least once before, refers to a particular building or group of buildings that houses a school. Do not use #2.

There's not much difference between 3 and 4, though. Using "class" could mean any class, while "the class" means a particular class that both the speaker and the listener know of.