No, to me, those are quite opposite. 'He is absent' has an implication that he is absent without leave. If he was away with approval, it would be more normal to say 'He is away from work'. Yes, I think that 'a leave of absence' works. Perhaps with a formal style like that, it would be more appropriate to use 'he applies' instead, to match the register.

A "leave" here means permission. A "leave of absence" is a fairly formal term. It means someone asked for time away from work (or other duty), and got permission. It means an authorized absence, for some unspecified reason. So-and-so "is absent" is a simple fact: that person is not present. It sometimes carries a suggestion that the person is absent without permission. A teacher takes attendance: "Joan Ellis?" "Here." "Randy Gomes?" "Here." "Michael Johnson? Michael Johnson? OK, Michael Johnson is absent." He might arrive later on with a permission slip, but he probably won't. At a meeting at work, on the other hand, someone might well say "Tom is absent today--he's onsite at the customer's--so we will proceed without him."

A leave of absence is longer than a day off. But you can have a day off from work and be absent. So in Q1 means that he was not at work. Q2 No they aren't the same thing. Q3 Yes. You can ask for a leave of absence.