Yes, perfect.

You could say, "They live near the school" or "They don't live far from the school."

"School" is singular and countable in your example, so it needs an article or some other determiner. "They live not far" has an overly formal tone." I would suggest: They live near the school.

Yes, this sounds perfectly natural.

A student can skip the ‘the’ but others normally wouldn’t. This is because a student who ‘goes to school’ is participating in an activity. Others are going to a location, which requires ‘the’.