You must go to a trading site to trade safely or you might get scammed.
Here's a couple, and you can find many more by searching Mafia Wars Trading groups mainly on facebook.
https://sites.Google.com/site/kingpintradingmarket/home
http://mwtradingclub.org
(this one also has a ratio guide if you decide to trade on your own or just wonder what the proper gift is to return if you don't have something the other person has on their gift list. Weapon Parts is always a good substitution, and this guide shows you how many WP are equal to different gifts)
Looted items in Mafia Wars helps you to win more fights if the looted items are better then your current items you already have.
You cannot gift some special items in Mafia Wars. You can check out my site for more guides on Mafia wars and other facebook games.
no
you will get rare and uncommon items.
Jobs
It's not available anymore, you an try to trade the items on the mafia wars forum. Go here: http://forums.zynga.com/forumdisplay.php?s=c9617812b8ed9e525f1dc589a2018aad&f=39 And go to what site you play it on. You can post a thread there requesting the items you need.
If you search mafia wars wiki, it is the most comprehensive mafia wars info site on the internet. They keep it very up to date
no you cannot delete them but if you want to get rid of the loot items, you can send them to your mafia family members.
Why yes, yes you do.
You keep them, but they are of no value. Trade with other players for other items you need. You can only vault one set for each.
You get looted items when you fight other mafias and win.
No there isn't any teddy bears on Mafia Wars as loot or vault items.