When i get online and get the patches for the 360, somehow, will i have to start a new character so I don't have the glitches, etc..

No, not at all, all updates will be working on any character as soon as you download the update, you still may experience a few minor glitches, but Bethesda are still battling against them whilst you enjoy the game ;D

there's a high chance that broken quests and items will remain that way if you don't start over

This is somewhat true, if you're having problems getting a certain quest to start don't expect to do it on that character, but aside from that everything should work fine.