Mudkat, are you sure the air comes out of your reservoir when you open the fill hole, or is it possible the air is rushing in? If you have a suction on the reservoir, that could cause a problem. If there is positive pressure, that should actually make the pump work better. It sounds to me like when you open the suction line, you are allowing air in to relieve the vacuum and then the system works. If you remove the fill plug, is there oil on the dipstick? How about leaving the cap off and operate the hydraulics? If there is a suction in the reservoir, the hydraulics should start to work if the fill cap is off. A vent could be plugged causing the vacuum to form when the tractor cools. As the tractor heats up, that vacuum would also decrease. Either way, a pump running dry will experience a quick death, so be very careful operating/troubleshooting this problem.
How about the hydrstatic transmission? Does the tractor have power? Both the transmission and the hydraulics use the same sump (and sump suction screen) and main filter on the right side of the tractor near the right-rear wheel. Then the fluid splits off and goes through the hydrostatic transmissions charge pump and then to the oil cooler and a separate hydraulic filter on the left side of the tractor.
I'm thinking that if the hydrostatic transmission works, your problem should not be the large filter or sump screen. Of course, if there is a problem in the hydrostatic transmission, that could be what's causing your vacuum at the hydraulic pump inlet line.
Have you talked to your dealer? Did you buy from a dealer or individual?
There are lots of things to check, but you have to keep some oil in that pump or you will soon need a pump whether that was the original problem or not. Good luck finding the problem.
I'm talking about a hydrostatic transmission because you said your tractor was a "35d" model. If it doesn't have a hydrostatic transmission, it's a TC35 without the "d".