"He is not here."
in my opinion, this is a little more formal than is common. Seems like the first three words would suffice
Yes, but the solution sentence extends with more hanzis. 他不在 is the short form of the same meanings. This is also correct.
The only possible problem with 他不在。 is that phrase is a euphemism for he is dead
It suffices in most cases, but still it depends on the context. e.g. If the sentence is used in clarifying that the subject is not here but would rather be in one or more other possible locations, then you need to mention 這裏.