Apparently I have unlocked some skins and other customization options. Can I apply these to my existing character or do I have to start a brand new character?
Any skins that you have unlocked/found while playing can be applied to your current, or any other character on your account.
You can change the look of your character at any time via the customization shop.
If you have found/unlocked additional skins for use (beyond the ones available initially), these can be found in your backpack. Click to use any customization items you wish to add to your list of selectable options in the store.
Customization items are class specific, but you can activate skins for use even if the class the item is for doesn't match your current character's class (i.e. I'm an Assassin but I can still activate a Gunzerker skin).