Unfortunately, I don't think you can
There is no college of illusion but there is the College of Winterhold. As the name suggests, it's in Winterhold (north of Windhelm).
You can sell it to Urag gro-Shub, the librarian at the College of Winterhold.
There are two factions with 'college' in their name, the College of Winterhold, and the Bards' College.
Yes you need to do the quests of the college of winterhold and they come to you.
It's a College of Winterhold quest. Just do those quests and you will stumble along the quest :)
Well I know Septimus is in a little cave/hut thing north of College of Winterhold.
You talk to the bridge guard and return to Skyrim. That is essentially the end of the main quest line. However, there are several other story arcs to be explored. The Thieves Guild, College of Winterhold, Dark Brotherhood, The Companions, The Skyrim Civil War. If you have the Dragonborn and Dawnguard DLC, you can also follow their main quest lines. And once you have completed everything that vanilla Skyrim has to offer, you can install mods like Moonpath to Elswyr, Falskaar, The Forgotten City, Project AHO, and Beyond Skyrim, some of which add entire regions to the game.
The Mages in the College of Winterhold sell staffs but if you mean an actual shop then some might have them (not quite sure)
The shrine of Azura can be found south of Winterhold
Skyrim for Pimps - 2011 Crotch Vs Winterhold 2-11 was released on: USA: 25 July 2012
The spell tome can be bought from Drevis Neloren at the college of Winterhold once the player reaches level 70 illusion.
Factions you can join: Companions College of Winterhold Dark Brotherhood Stormcloaks Imperials With the Dawnguard DLC: Volkihar Vampires Dawnguard