What is the tire pressure for a 2004 Yamaha YZ 250

The factory recommends 15PSI. I usually run 12PSI.

Depends a lot on what tires you have and what kind of "track" you are riding. You typically want a firm tire (more pressure) for a soft surface and a softer tire (less pressure) for a hard track but there are always exceptions depending on the details of the track and what tire you have mounted up. If you are riding a track with a lot of hard braking bumps or firm ruts or even tree roots (if you ride trails) you should run more pressure - like maybe 13-15PSI. If you run less than about 10PSI you risk pinching the tube and getting the rim damaged from sand intrusion between the bead and the tire which can cause a lot of problems unless you change your tires frequently and run multiple bead locks. To run really low pressures usually means you don't have the best tire on your bike for the conditions in my opinion.

To simplify:
Soft track, hard tire - run 12-15PSI
Hard track, soft tire - run no less than 10PSI

The service manual just says that if it leaks any oil to replace it.

replace your clutch plates... if you have already done so, you may have assembled the basket, plates & pressure plate incorrectly.

Sounds like dirt in the carb, there are two sets of jets, one for idle one for load.
Strip and clean the carb.