You will need to add a 7 wire round plug to your tow vehicle. Best talk to a Toyota dealer mechanic as there is most likely a kit you can purchase to do this. It might be best to let the dealer do it but, that's up to you.
I like my Prodigy P3 brake controller. After you get the above 7 wire kit installed you can purchase a wiring kit with the brake controller that will allow you to plug it right in to the wires under the dash of your truck. After that your trailer brakes will work.
I have no idea on the wheel spacers.
With the battery installed and the propane tanks filled and turned on most of your TT systems should work. The only things that wont work are the things that run off of 110 volts. Those being the roof mount AC (if you have one) and the outlets inside the camper. If you want them to work you have to plug the trailer in. You will have to post the make and model of your fridge and water heater to get directions on lighting those. There are different ways for different units. If you get the make and model and do a internet search you might even get lucky and find the owners manuals on-line.