I think the weight carrying numbers of 5000 / 500 are ridiculously conservative. Any domestic 1/2 ton truck from the last 50 years will be just fine with your car and trailer without a weight distributing hitch. Tongue weight should be 10 to 15 percent. Too much is a lesser evil than too little. 1000 pounds would be good. You may not be able to get exactly that tongue weight depending on the deck length of the trailer and location of tie downs. I.E. how far fore and aft can you move the car and still chain or strap it.
Chain or strap to the wheels, axle, or lower control arms. You want the body and suspension of the car to be able to move. If you were to chain to the frame on the car, you would never get it tight enough, and every bump and dip would loosen/tighten your load securement.
If the rear suspension on the truck does not look overly squatted, then you are good.
Air up all 10 tires (truck, trailer, and spares) to the max on the sidewall.