Typically it is more expensive to trade in your car to a dealership, I.E. you get more money is you sell it privately. The advantage to trading it in is you don't have to worry about selling it and its instant.
And yes the $6,000 you owe would be subtract from what he gives you. If he gives you $12,000 your down payment would effectively be $6,000.
Typically the dealership has financing options and you can get financed right there. If you want you can go through your own bank.
As far as AT/MT go, what do you have now? In a 335 coupe, I would highly recommend MT but I'm not you!
Here are two links - the first one is Kelly Blue Book so you can see what your current car is worth. The second is an auto loan calculator that simple to use to you can setimate payments: