Is it absolutely necessary to do his quests to become the arch-mage? My game glitched up, I can't complete his quests.

Once you get to Wizard level, you should be able to challenge Trebonius directly. You may have to do all the Ald'Ruhn quests though, not absolutely sure about that.

So... I could completely skip the quests in Sardith Mora, given by Skink-in-tree?

There`s three ways to become the archmagister of the mages guild.

1. Complete Skink-in-Tree`s quests. In this way both you and Trebonius will be archmagisters at the same time. You won`t get his amulet of his staff.

2. Challenge archmage Trebonius for his title. You`ll fight him in the arena and get his amulet, staff and status once you kill him.

3. Complete archmage Trebonius`s quests Mystery of the Dwarves and Kill Telvanni councilors (Be sure you do all Telvanni quests before you do this one or else you`ll never be able to do much in Great House Telvanni). If you complete his quests , Trebonius will be impressed with you and reward you an advancement to archmage and give you his staff and the Necromancer's Amulet.

You can do Skink's quests without becoming archmage. Just don't ask for advancement.

That's the opposite of what I want to do. I want to become archmage without doing skink's quests.

Yes, I know. And yes, it's possible. Just skip to Trebonius after doing Edwinna's quests. Lissenber seemed to imply that doing Skink's quests will result in never getting rid of Trebonius, which isn't necessarily true.

I've never done it that way personally, but according to the UESP, it's possible. I have done it after doing only 2 quests for Skink-in-tree, however, so you definitely don't have to do all of them.

Nope. Perhaps I was a bit unclear. You can still kill Trebonius but this doesn`t involve becoming the archmage through Skinks.