2 big problems with 2002 GT

[Technick]

For the starting problem you have either a plugged fuel filter or the fuel pump is weak or both. Try cycling the key from off to run a couple of times before going to the crank mode. You should hear the fuel pump cycle on for 3 seconds and then go off each time you cycle the key. If cycling the key helps then chances are the by-pass in the fuel pump is bad and the system is not holding pressure, or is bleeding off pressure during key on engine cranking mode. These newer stangs, (99 up) have a returnless fuel system, in other words, there is no return line, just a feed line. Also did you know that when the engine is running that the fuel pump does not run continuesly? It cycles on and off in it's search for the proper fuel pressure to the injectors. Only at WOT is the pump usually running continuesly.
The bucking back and forth of the trans sounds like it could be excessive backlash in the trans, or the rear end or both.
Both of these problems should be taken care of under warrenty.