I have a 2002 VW Jetta, it has a 2.0 motor. I've

It could be loose or broken motor/transmission mounts. Since it only does it on acceleration, I don't think the tires would be the problem.

Well if the check engine light is on then there is something wrong, now you need the engine control module scanned for fault codes to begin to understand the problem that turned the light on, you need an OBD2 code reader (about $35.00) as a minimum to read the codes.

You have a random misfire issue follow the link for info on this problem. Problem 6 Engine Stalls Misfire Code Cylinder Misfire

Sounds like the chip in the key is not allowing the security system to allow the car to start. You may need a new key from the dealer.

Most VW are interferance motors. That would mean the pistons could have hit some valves. I would stay away.
Also have you seen the motor? I may be off but I thought the VR6's had a chain. Even if, its still an interfernce motor.

YOU HAVE 3 MOUNTS. YOU HAVE PASSENGER SIDE ENGINE MOUNT. YOU HAVE DRIVER SIDE ENGINE MOUNT AND THE LOWER TORQUE STRUT MOUNT TO TRANSAXLE.

did you check compression cilinder?maybe you have the pocelanite spark plug craked or the rubber coil protection damage

Sounds like its missing on 1 or more cylinders. A check of the plugs and wires is in order, Could be a fouled plug or a loose/bad sparkplug wire. Best of luck.
Greg

Have you checked your transmission fluid?

Your car overheating is either low coolant or its the thermostat. The thermostat is what tells the car to add coolant and turn on fans.
Your tranny slips could be because low transmission fluid, which was the cause with my VW. Or your tranny is bad, which is common with VW.
When your car shakes when you accelerate does it seem like its choking?