My car shakes violently when in drive, the entire motor moves from side to side and is not driveable, but it is fine when it is in reverse. Whats wrong with it?

I am thinking that with the back firing ans all ,you put the ht leads on in the wrong firing order and had 2 wires crossed
when replacing the new with the old you fixed the cross wires but reintroduced a bad plug or a bad ht lead.

Sounds to me like the splines in the transaxle are shot

Unbalanced wheels shake a vehicle like that. It's hard to understand your earlier statement. It sounded like after you weren't moving anymore that it still shook. I am figuring that isn't what you meant.

Is it an automatic or manual trans ?
An automatic should downshift when you push on the gas pedal. The shaking could be a miss-fire or lack of fuel. The noise could be a knock from the engine or something in the exhaust system.
The freaking jet noise could be an exhaust leak or something going on with the air cleaner or a vac leak.

check you motor and transmission mounts. if one of them are wore out it will cause the car to do this. i just fixed a mazda with the exact some problem yesterday, but the situation was in reverse. runs like a champ now

Probably one or more bad motor or transmission mounts. You can brake torque the engine in reverse and drive, if the engine rocks violently in either step those mounts need replaced.

is it shaking due to rough running ? you may need a tuneup if not you may have a broken motor mount, with the hood open and the front and back of the vehicle is clear (not ruining anything over) stand to one side of the vehicle, observe the engine and have a helper with there foot firmly on the brake put it into reverse and apply the gas gently, then put it into drive and repeat the engine should move very little if it moves considerably replace the motor mount

It could be tires need balanced, but more likely it's one of your cv joints. That is basically a "drive shaft" on front wheel drive vehicles.

Check the ball joints and cv axles this could cause the car to shake violently and most tire stores will check them for you for free.

Without actually being there, it could be a shock that has failed. Normally to check that press down on a corner of the car (and or try to get it to bounce) then let go. If it continues to bounce like a ball then that's the corner that has a bad shock.
It's somewhat comical if you see it happen when driving. Also have someone follow you while driving over an area that you know this will happen. They may see it happening.