1. He is absent from work today.
  2. He absented himself from work today.
  3. He is absent today, he never came to work.
    Please correct all.
  1. He is absent today, for he has not come to work (because he did not come to work).

No corrections were needed.
3. He is absent today. - this would usually be enough, however, if you wanted to emphasise the fact that he has been absent all day, and not just for a part of it, and you have not been given a reason for his absence (such as illness) there could be a case for the way you have written #3.

Anglophile’s suggestion is awkward and redundant. There is no point in saying he is absent because he hasn’t come to work. If he hasn’t come to work then he MUST be absent.

Good logic! Agreed. But can it be awkward and redundant in all contexts?
Let’s take this hypothetical situation:
Why do (How can) you say he is absent?
(I say) he is absent because he hasn’t come to work today.

Yes, that will remain a corollary even if repeated!