Okay so, I've already completed the main quest on a thief/archer type character, focusing on the main quest using fast travel and not focusing on RPing too much.

I created a warrior type, specializing in 2h, heavy armor, alchemy, alteration, and restoration. On the last quest for the Companions (I won't spoil anything) he visited Winterhold and learned about the College of Magic there, and he became interested because he wants to further his understanding of the alteration and restoration schools. For those who have completed the Mage College storyline (WITHOUT spoiling anything!) is it possible for my character to complete it without using destruction, illusion, or conjuration? Would I be reprimanded for using a 2h weapon for most of the quests? Any help would be greatly appreciated.

You can pretty much get away with just about anything in this game. I don't know about the "without using"... because that could mean "EVER using" or simply "not often using". There's no strict prohibition on using a spell that you're not good at.

Do what you want 'cause a gamer is free, you're a gamer!

Well I guess I mean not often using. Like I just want to know if there's a restriction on weaponry while doing the quests for the college, or something similar.