I am confused. There is obviously some history here that you have not told us. You found that the fuel pump relay had been removed and the "hydraulic pump relay" is missing (I don't even know what that is on an Audi unless it is the ABS pump in which case that explains the ESP light). Who was poking around and removing relays and why? Can you tell us how you got to this point?

I am not sure what you mean by "car goes back and forth with gear changes" but that it does not accelerate. Does this mean you get a little forward and backward rocking when you put it in gear but if you step on the gas the engine revs but you do not go anywhere? If that is the case there is a problem with the tranny or the TCU. Did you try putting the car in neutral and seeing if it rolls? If the ABS is messed up are the brakes locked on?

Get a VAG-COM and read the codes. You probably have a million of them. You then have to sort through them and see what you got.