Wheelsdude: Welcome aboard!
1. Handi's are the trucks, plain jane wood ect., while the ultras, many times have laminated stocks, muzzle breaks, and better finish.
2. .500 S&W would be a thumper, while I've no direct experience with one, but anytime you move a slug that big that fast it'll boot you. Besides one of the beauties of these rifles is that they are light.
Remember your talking to a fellow who uses a 12 ga.Tracker II for deer and a Pardner 28" with screw in tubes for turkey. Either will rock your world, but these aren't plinkers, recoil and ammo cost {Win. Supreme 1 5/8 oz. 5's, Brenneke KO 1 oz. sabots} preclude that, you sight them in and hunt.
3.All the calibers you mentioned would be excellent choices, right now one of my favorites is a used Handi in .270 I bought at Jay's in Clare. Don't forget that you can taylor a Handi by buying additional barrels. What this allows is that you can have a deer rifle and buy a Hornet or .223 barrel for varmite. Then you have is a multiple use rifle, that you become very familiar with.
Good Luck:
<>< Duce: