Can I store stuff in the chest in the Archmage quarters without it dissappearing? If that is true then why would I ever need a house?

Yes it is safe to store, however, you may want a house as you might find it inconvenient to keep traveling back and forth to winterhold just to store your stuff.

That perks comes to the leader of all guilds. I prefer a house because I can customize it, get kids easier to travel to plus it's in a real city. I hated the AM quarters because it took so long to get there from fast travelling to the college you have to go through 3 loading screens, one of wish is a long one

I haven't been back there in so long, and I don't have a house..

Quite frankly, having more than one house is a pain as I never remember where my stuff is.

For the record, the entire Arch Mages quarters are safe to store stuff in.

The only reason you have to buy any house in this game is to become Thane of a Hold (buying a house is one of the requirements)

Completely secure, far as I know.
Additional benefit that houses provide:
* Well Rested bonus (AQ only gives Rested)
* Ability to move spouse there (LC bonus).
* Ability to adopt with HF, with all perks/responsibility inherent.