You are correct that if you have very poor compression, the engine will vibrate some...quite a bit if the compression is terribly low. For those of you who would love to berate this next comment, I capitalize the important: PART of the reason is that a cold engine is not filled with as many expanded parts as a hot engine; thus, more escapes are possible for fluids and gases.

I would most definitely run some injector cleaner through your car. Red Line is the compound that I use. No more expensive than the others, and you're getting the same freakin' product. You should use the recommended amount of the bottle every fuel up. Proper maintenance is a key in cheaper repairs.

While you have your head under the bonnet, take off the throttle body (some of it may require the installation of new gaskets, so be aware), and clean out any residue. When the engine idle drops as I am understanding from your statement, the vibration gets worse. Yes? This would make sense, as idle is not the smoothest/fuel economic position your car has available. I am thinking that maybe you have a leak in a vacuum line somewhere. Check all of your hoses that connect to your throttle, engine, PCV, air intake systems, etc. A small leak can lead to improper idling and an incorrect air/petrol mixture.

Might not solve your problem, but hopefully you can reduce your options as to the what the problem might be.