Before you start doing any filing of the nut - have you got a floating trem? if you do I have feeling that this may be causing some of the problems. [font=Arial, Helvetica, sans-serif]The vintage and US Standard (two-post) trems are designed to float with their front edge down and their rear edge up, with approximately 3/32" gap between the bridge and the body at the rear edge. You will need to lock your trem [/font]in place while you perform other adjustments.
First, loosen the two screws that hold the claw in the rear of the guitar, loosening its screws will allow the bridge's rear edge to rise. You don't need to loosen them to the point where the screws fall out, but loosen them several turns so the rear edge of the bridge is higher than it should be. Now, in the rear of the guitar, look for the gap between the trem block (the metal block that the springs plug into) and the rear wall of the trem cavity (the wall closest to the butt of the guitar), and push a small wedge into it. This wedge is something you can make out of wood; it should be about 1/2" wide, 1.5" long, and roughly 1/8" to 3/8" in thickness. As you push the wedge into the gap, you'll see that it forces the rear edge of the trem back down towards the top of the guitar. Push it in until the bridge is positioned in its proper floating position.
Be sure to push the wedge in at the center of the block, that way you'll be able to use the screwdriver a little later to adjust the two claw screws without moving the wedge. Now, with the wedge in place, you'll see that you can tune the guitar to pitch without the bridge pulling up and set your bridge height/saddles and intonation. Be careful doing the rest of the adjustments that you don't dislodge the wedge as it will be sticking out of the back of the guitar. Take care when you hold the guitar, and especially if you lay it down on a bench or table.
Hopefully that should be the problem sorted out for you