Hey how about that, it's a question I can help with!
X1 on making the trailer level. I've had them be low in the front by an inch or so with no detrimental handling but the front 7" higher than the back will lead to swaying and even worse, poor control and braking. In an extreme braking situation it can lead to a jackknife. The trailer height should be determined when loaded. The ball height on the truck will sink several inches with the car on the trailer as opposed to an empty trailer. I'm guessing the gross weight on your empty trailer is about 1500 lbs, and a 66 Dart should be about 2800 lbs for total payload of 4300. So a tongue weight of about 375-450 lbs is about right. Taking the rig to a commercial truck scale is well worth the time and the few bucks it will cost to get an accurate weight on each axle with your loaded trailer. Load distribution is important, as you don't want the car too far forward or rearward on the trailer. About 55-60% of the load in front of the midpoint between your axles is a good number. Don't worry about your tires getting overheated or worn, if they are properly rated for the load and properly inflated they will be fine. At the commercial scale you can move the car forward and backward a bit to get a true reading of the tongue weight and load distribution. Once you find the sweet spot mark the trailer floor or frame where the front tire sits and you won't have to monkey with it again.
Also, make sure you look at the hitch on your truck, it will list the max tongue weight and total weight it will handle. Most any half ton truck should be able to tow your combination well.
Not to nitpick, but the receiver on your truck is the 2" square receptacle mounted to the frame. What you currently have is a 3.5" drop on your stinger (the part that slides into the receiver). I am a huge fan of adjustable stingers, as it's a snap to raise and lower it for loaded or empty trailers, as well as taller/shorter trailers and different gross weight trailers. Below is mine, and the two parts to the right are an Adjust-A-Ball system. That ball system has one mounting shaft with interchangeable balls covering the three most popular ball sizes (1-7/8", 2", and 2-5/16"). With this combo I can tow anything within the capacity of my truck.
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When it comes to securing the car, I've tried every method, and correctly done all will work well. As RogerRamRod says, the over-the-wheel strap rigs work well and are very adjustable. Whether you use one each on the front tires with chains on the rear axle, or use four of them (one per wheel) they work great. The straps and how they are arranged and tightened can be a bit confusing at first but once you figure it out they work fine. Securing via the frame of the vehicle is a more solid arrangement since the car won't be moving around on the suspension but unless you're on very rough roads or driving like a bat out of hell, the wheel nets should suffice.
Setting the trailer brakes needs to be done for both empty and loaded as they will be very different settings. For either configuration, find a wide open parking lot and get the vehicle going about 10 MPH and pull the manual brake lever all the way on. The brakes should just lock up and start sliding. This is a good starting point. Now drive at about 35 MPH, and brake using just the foot pedal. You should feel the trailer brakes grab just before the truck brakes. Adjust the setting as needed. Especially when loaded, I like the trailer to start braking JUST BEFORE the truck starts braking. You want the trailer to brake just a hair (and this is truly just a hair) more than the truck. Once you find the right settings jot them down for both empty and loaded, for future reference.
One last thought, the weight of your loaded trailer is going to be fairly close to your truck weight. On any kind of uneven road this can lead to bucking as the trailer is going up while the truck is going down, or vice versa. It's very tiring and can be eliminated with an equalizer hitch.
If all of this is clear as mud feel free to send me a PM and I can walk you though any needed details.
Zach