Speedmeter is bouncing up and down. Then will stop for a while then continues, then stops. OBD2 reader put out code P0503

All the things your having problems with are controlled by the ( FEM ) front electronic module . The thing to do would be check for DTC'S - diagnostic trouble codes an this means taking it to a ford dealer or some other repair facility with a professional grade scan tool , a OBD2 code reader won't do it.

Looks like it could be a "System Time Plausibility"
http://engine-codes.com/p1515_bmw.html

or Throttle System Throttle Response
http://engine-codes.com/qa/1755/what-is-bmw-code-p1515

run fault codes and if automatic gear box look for a faulty TCM that is not changing down gears as you stop.

Havbe the engine electonics been tested for fault codes? Buy a basic code reader for the OBD2 control system and test it.

well first check if they leak. if they leak they will have oily film on them. then if they dont leak u can do bounce test put ur knee on the rear bumper and bounce the car up n down as hard as posible the wen u stop if the car continues to bounce more then 3 times u need shocks

Put a obd2 scanner, code reader on it. It will tell you.

OBD2 codes could stop starting. Buy a OBD2 reader from an auto parts store, ($70). Read the codes and write them down. Clear the codes. See if will start. If you need to find the OBD2 port look in the repair manual CD. ($16 Ebay). On a 2003 TDI it is under the steering wheel, just to the left a bit. If you have codes, you can look them up on the repair manual CD.

Does it seem to turn over quickly? If so i'd suspect the timing belt broke considering the rattling noise heard before it stopped.

I have a 2003 new beetle TDI. I have got error codes on my ECM that have stopped my engine from starting. I got an OBD2 reader from the auto parts store. I plugged it into the OBD2 port below the steering wheel and got the error codes. After writing them down, I cleared the codes. the car started fine. It has pulled this a couple times over three years. I carry an OBD2 reader with me so I can clear the codes and get going again. Good Luck

MAKE SURE YOUR ENGINE IS WELL TUNED. MAKE SURE TURBO IS WORKING NO RESTRICTED EXHAUST..ALL PLUGS AND WIRES ARE GOOD.SOUND LIKE TRANSMISSION SLIPPING.HAVE TRANSMISSION PRESSURE TESTED.IF FLUID LEVEL GOOD AND TRANSMISSION FILTER NOT STOPPED UP.PROBLEM COULD BE TRANSMISSION SHIFT VALVES OR TRANSMISSION PUMP OUT PUT PRESSURE TOO LOW.