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You have to become a member of the mages guild. For you to do that You MUST do a series of misions.
longsword, nowhere, but most mages guild halls have one in the open somewhere
many traders will sell some lesser effect scrolls, the mages guild is free to join and will provide you with many spells and scrolls and when you enter an oblivion gate many deadra will drop useful scrolls
the mages guild halls usually have someone who can recharge your weapons. but it can be expensive. you can also find varla stones in Ayleid ruins that will recharge all the enchanted weapons in your inventory.
i am sry but the only way to get into that building is to go through the mages quilds and have the guy open it for you once you join the university
Yes actually you can, there is a magic spell called "Levitation" you can buy it at some of the mages guilds, I believe.
Yes Such as, The Mages Guild The Fighters Guild And several other things such as the Imperial Legion or Imperial Cult and many more.
If you've been kicked out once before, and re-accepted, there is no way back in if you got kicked out a second time.
Assuming you mean spell scrolls, both merchants at the Mystic Emporium in the Imperial City Market District sell some. Check the Mages Guild branches in each city, they have scroll selling merchants too.
The mages guild in skingrad
In The Elder Scrolls IV: Oblivion, you can find five trainers total. The three basic level trainers are Esbern in Cheydinhal's Great Chapel of Arkay (Knight's of The Nine character only), Hil the Tall in Cheydinhal's Great Chapel of Arkay, and Jantus Brolus at Istirus Brolus' house in Bruma. The one advanced trainer of Illusion is Carahil at the Anvil Mages Guild. And the only Master Trainer of Illusion, Martina Floria, can be found in the Arcane University's Chironasium.
You can find people who sell spells in the Market District of the Imperial City. they can also be found in the Mage Guilds, mostly all of the mages sell spells except for the alchemists. Mage Guilds are found in Chorrol, Anvil, Bravil, Bruma, Cheydinhal, Leyawin, and Skingrad. You can also find and craft spells at the Arcane University. The spells that are sold at the University are more for advanced mages with lots of money and mana.