Re: 5th wheel/gooseneck tow rating
Weigh the truck after you fill it with diesel, including driver, passenger(s), toolbox full of tools and extra fluids, cooler full of cool, and whatever is normally in the pickup when on the road. Subtract that weight from 20,000 pounds GCWR and that is the max gross weight of any trailer you can tow without exceeding the GCWR of the tow vehicle.
But if you are talking about a bumper-pull trailer, the receiver hitch is often the limiter. The receiver has a weight-carrying (WC) and a weight-distributed (WD) rating stamped on the receiver. For an '01 it was probably 500/5,000 WC and 1,000/10,000 WD. (The first number is the max hitch weight and the second number is the max gross weight of the trailer.
If your F-350 has single rear wheels (SRW), then your limiter for a gooseneck or 5er trailer is probably GVWR. Your GVWR is 9,700 or 9,900. Subtract the weight of the wet and loaded tow vehicle (including installed hitch) from the GVWR, and that will tell you the max hitch weight you can tow without exceeding the GVWR of the tow vehicle.
If your F-350 has dual rear wheels (DRW), then you don't need to worry about exceding the GVWR, so just watch the GCWR for a 5er or goose, or the receiver rating for a tag trailer.
The Ford tow rating for a conventional (tag or bumper-pull) trailer is limited by the Ford receiver. You can get rid of that limit by replacing the factory receiver with a Reese TowBeast Class 5 hitch that is rated for a lot more weight. So then the GCWR or maybe the GVWR becomes the limiter for a tag trailer