Hello, Organbuilder from Sweden,

Q1.Is it then meant that the front torque tube bushing is meant to center the driveshaft in the torque tube and prevent the shaft from displacing itself laterally?
A1. I would say the the front torque tube bushing is meant to keep the Rear Yoke centered inside the torque tube.
Front Yoke on left, Rear Yoke on right
The rear torque tube bushing is meant to center the driveshaft in the torque tube.

Q2.If the front bushing is shot does that mean it is time to replace the rear one, too.
A2. I would say not necessarily, but since the front bushing is shot, you should replace all four pieces.
Front bushing
Rear bushing
Oil seal
u-joint/trunnion
Don't forget U-Joint Bolt Lock Clips

You can install a Okie Bushing - Driveshaft Bushing/Seal Assembly
Assembly Here
Home Page
Okie Bushing Photo Album Instructions on 2nd page of album.
If you have a dowel pin holding in your front bushing you don't need the front bushing puller.
Just drill out the dowel pin and fabricate a hook to pull out the front bushing.
The old rear bushing and oil seal stays inside torque tube.

The Okie bushing was created by the long gone National Machine Works Inc in Oklahoma, hence the name Okie.
It shortened the time and effort to repair the torque tube bushings and oil seal.
If you did it the way GM did it, you would remove the entire torque tube from the rear axle.
Remove ring & pinion gears
Pull out the driveshaft from the torque tube.
Use special tools to remove and replace the bushings and oil seal.
Then reassemble everything. Pain in the Butt.
With the Okie bushing you just pull out the front bushing.
Then from the front of the torque tube you push in the Okie bushing sleeve, which has the Front bushing,
Rear bushing & Oil seal inside the sleeve. Less Work.

Q3.I can feel that the drive shaft itself can move pretty freely side to side inside of the torque tube.
Does that mean that my front bushing is faulty or completely gone?
A3.If you mean you can move the rear yoke freely side to side inside the torque tube.
Yes, the front bushing is faulty.
If you slide the rear yoke out of the torque tube and the driveshaft moves freely side to side.
The rear bushing is faulty also, and probably the oil seal too.

If you have that much side to side movement, it is pretty much a sure bet that is where the vibration is coming from.

Q4. Hey, am I in the wrong forum?
A4. Technically Yes, but the the vibration could be in the Engine or the Driveline.
We don't know, we are still Troubleshooting the problem.
Your question could be posted in ether:
The Engine Shop forum
OR
Driveline forum
That is up to the forum Moderator to decide.