Engine shakes/moves severely at idle, like it is not even mounted in the car !!!

I had a mechanic tell me once he had an engine that was shaking like a dog passing a peach pit.  Love that analogy.

I'd say you have some uneven combustion.  I'd start with a basic tune up.  Check the carb sync if you have dual carbs.  Pull the plugs and look for a plug (or two) that is wet or covered in carbon.  Do a compression check next.

If your engine's moving up and down that much, then your front transmission mount is shot. And if the two rear transmission mounts aren't gone as well, I'd be surprised. Bad transmission mounts can cause the engine to pull on the clutch cable and/or throttle cable causing surging/shuddering as well as screw up the shifter linkage alignment making it hard to shift. You'll also want to check the clutch cable guide (bowden tube) between the frame and clutch arm on the trans has at least 1 inch of sag in it to provide the flex needed when things move around.

That said, your engine shouldn't be bucking that much at idle. I've ran my engine just sitting on the floor before and it'll only really move if you rev the engine. Definitely need to fix this as well.

You should probably get yourself a Bentley manual to match the year of your chassis. It'll give you most of the info you need. Otherwise, doing a search in the forums over on thesamba.com should cover most of the rest.

Sassy  --  pretty sure Justin nailed it for you:  You probably have broken mounts.  Don't drive it until you replace them!  You will need to pull the engine in order to replace the mounts.  Other than that, the mounts are easy to remove/replace (two bolts each).

Your situation is obvious and self explanatory just in your mention of how the engine visibly moves around (with intact mounts, movement is barely perceptible).  As Justin pointed out, the combination of your other symptoms point directly to broken mounts: Erratic clutch actuation, inconsistent gear shift selection, "interesting" throttle pick-up from off-idle when in gear.  All of the linkage for clutch, shifter, and throttle rely explicitly on the engine/transmission staying fixed in one place. 

You might also have other problems that cause rough running, but before you look at anything else:  Check those mounts.

PS:  I have had broken mounts  --  it is exactly as you've described.  Another pointer:  Do Not buy the red urethane mounts!  You are better off with the OEM rubber mounts or the HD mounts from CB.  The urethane mounts will shatter.  (I've had a chance to experience broken mounts twice . . . . there was the first time and then there was the second time when I replaced the urethane mounts I installed the first time).

Sassy  --  and another thing:  If you end up replacing the motor mounts on the transaxle bell housing, then you absolutely should replace the clutch throwout bearing too.

You should take advantage of the opportunity to replace the throwout bearing ANY time you can can see and touch it.  And if your clutch has unknown heritage, replace it and the pressure plate too!  I'm talking about only $100 in parts altogether.  So don't put that engine back on the trans without taking this "lucky" chance to replace them.

Yes, it looks like it's time to pull that old motor. and replace the engine/tranny mounts. 

As mentioned in an above post, go with stock mounts, not red urethane.

If you are going to drive it agressively I would add a mid mount such as this,

http://www.vwispwest.com/MB17-2536.html

 It will save your  tranny mount and help keep your tranny shift linkage from coming out of gear under a launch condition, besides that its cheap and easy while you have the transmission out . Mike McCarthy