Need an opinion/diagnosis. Eng./Tran./Clutch issue

Basically, the car has a noticeable vibration in neutral (5 sp. manual). And no, it's not the normal vibration from the V8. But, when you push in the clutch and put it in 1st gear and keep holding in the clutch, the vibration goes away. Also, the car stumbles/shakes pretty bad when trying to let out the clutch smoothly in 1st gear from a stop.
 
I would start with looking at the plug wires, and plugs. I nearly rebuilt an A4ld in an Explorer due to this type of thing.

Then look at things like the balancer to see if it slipped. However, when it does, it will usually shake all the time.

Then its time to check the flywheel, bolts, etc.

As you describe it going away with use of the clutch pedal, it may be the pilot bearing, the throughout bearing, or the front bearing/retainer in the trans, since that spins in neutral.