"There is a piece of steel that looks like a leaf spring from the top of the top link down to the rear and connected to a short piece of chain. Should that be tight or loose when mowing??"
Boy, I wish you had a picture of that top link set up!
During mowing operations, your top link should be pretty long to allow as much travel as possible to follow the terrain. Since you have a chain rigged, this is probably to limit how low you can drop the front of the cutter. My 790 has an adjustable stop on the rock shaft lever (controls the 3 point hitch) that limits how far the front drops.
Now, does your hitch have a chain instead of a top link? I'll assume you have a rigid top link and the chain is to limit travel.
For mowing, lengthen the top link as much as you can. As mentioned, this allows the cutter to follow the ground contours as much as possible. With the top link lengthened, the cutter will pivot up and down around the lower links.
To raise the cutter for transport (totally off the ground as if you were going to drive down a road with the cutter raised up), you'll want to shorten that top link as much as you can.
"Should the mower be run resting on the ground or lifted up??"
The tail wheel should be on the ground and the front of the cutter just slightly lower then the aft end. I suggest you just lower it until that chain is taut and see what it looks like. The sides of the cutter should not be resting on the ground. If the sides are on the ground, the chain is too long. Either adjust the chain length or use your rock shaft lever to raise or lower the front of the cutter until the front is a bit lower then the aft end.
Engaging the PTO
Your transmission should be in neutral. When you engage the PTO, set your throttle at about 1200 RPM or so, then engage the PTO. The cutter will vibrate and make some racket as the blades pivot out. Let the tractor run at 1200 or so RPM for 5-10 seconds, then smoothly increase engine speed to PTO RPM. Then engage the transmission in a low gear. You don't want to go too fast on your first use of a rotary cutter.
Now, some folks don't use a rigid upper link when cutting. They'll use just a chain to limit how far the front of the cutter can drop. I normally use the rigid top link, but sometimes I just use the lower links and set the front with the rock shaft (3PH) lever. I don't usually raise my cutter for transporting. I usually tow it with the tail wheel on the ground...but I don't run on pavement.
If you have a seatbelt and ROPS, make sure you are belted in. Depending on what you're cutting, you may want a mask and goggles to keep the dust and pollen out of your eyes and lungs. You'll definitely want ear plugs or muffs.
Watch your tractor's temperature gage. All that dust and stuff can clog your radiator. You may want to clean the grill and screen (at the radiator) at regular intervals when brush cutting.
Try not to fall off your tractor. that cutter will do as good a job on you as it will on the brush you're cutting.
I hope this helps... A picture of your upper link arrangement would be a great help for us to assist you.