whining sound is possibly a bad fuel pump, but why dose it stop after few seconds of starting? =S really odd.. that squeaking sound could be the clutch from the AC? heard you hit one of the AC buttons in the Vid. pretty sure its not normal for it to squeak(mines dosent). Z-car.net has alot of info on ruff starting/idle and just might help with your problem. i would think it has to do with fuel delivery on your Z....
-try changing/checking(if you haven't): fuel filter CAS(spin it to see if injectors work and fuel pump pumps fuel) Clean your throttle bodies/intake plenum try dumping one can of sea foam into your tank and let the engine have a little sip of it also.
[ http://www.z-car.net ] >tech >Curing Erratic Idle(do all this just to be sure, ALL) (= best of luck! nice Z btw! -Corey