I have no idea what rear connector came with the type package, but 6 pin seems odd to me as the 7 pin has been a standard for a long time. But it's pretty easy to get whatever pin count you need, so I shouldn't worry much.
You will need a brake controller if the trailer has brakes, the easiest way to tell aside from reading it from the info card when you buy, is to look for a blue and sometimes white wire running to the wheels on the trailer. If your trailer is light and short enough not to have this, don't worry about the brake controller.
If you do need a brake controller, spend the extra money on the adapter, there is a trailer tow connector under the dash, it's blue I think and has 3 or 4 pins. This will allow you to plug and play with most brake controllers without needing to splice into anything or rig up fuses under the hood.
The second cooler is a trans cooler, it's a nice big one, so as long as the fluid is a nice color, you're all set in that department.
If you haven't, check the mounting on the trailer hitch to make sure all the fasteners are tight and in good condition. If any are broken, you need to replace them, and if suggest replacing all of them because if one went the others are just as old. If you are unsure how to tell if you have a weight distribution ***** receiver and are uncomfortable making that determination, goto any camper store and they should be able to tell you in a couple seconds also any place that installs them should know. It's important to not put a weight distribution setup on a receiver that isn't rated for it. They aren't designed to take the torsional load the weight distributing bars place on it and can break loose if they fail. The safety chains won't do anything and the trailer brakes won't go into emergency.
If you are new to trailering, just ask for tips when you get your trailer, and browse around here in the trailer forum, plenty of great tips both for general towing as well as tips specific to our trucks.
Congrats man, enjoy yourself, you should be just about set.
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