Discussion of the Academy faction (Mages and Wizards). Tactics and their specific features.

1) If a Wizard casts a counterspell against a Necromancer, does the Necromancer get pushed back half a turn, or does the Wizard get pushed back a full turn?

I don't know, it needs to be tested. In my understanding, the Necromancer is not pushed back; the spell simply doesn't work. But there's still a question: will the Necromancer's "Wizard" skill work in this case, or not... I can't say.

2) Does the counterspell work on the "Scream of Horror" spell?

I have no idea. Probably, it should.

3) If Hafiz is a mentor and trains Nazir before he reaches the "Mentor" and "Arcane Knowledge" skills, and then Hafiz gains the "Scholar" skill while Nazir does not, will Hafiz gain the spells that Nazir has but Hafiz doesn't?

I don't think so. In the skillwheel, there's a note in the description of the "Scholar" skill that it doesn't apply to spells learned through artifacts. As I understand it, "Arcane Knowledge" has a similar mechanic.

4) If Hafiz casts a counterspell, and on the next turn, he casts a spell, and then the Necromancer moves, will the Necromancer's spell be blocked, or not?

I don't think so. It's better to do these things in reverse order.
In general, I advise you to test each of these points yourself. It's easy to do: in the map editor, place Nazir and Hafiz as the first player, and some Necromancer as the second player. Set everything up, run a staged battle (player versus player on one computer)... and you'll know for sure.