They will run a bit rough at cold start and with current engines this is largely due to the need to meet cold start emissions requirements. During initial cold engine operation, in addition to high idle, the mixture is enriched and timing is retarded to supply the front close coupled cats with a fair amount of hydrocarbons to light them off and get them hot enough that emissions are within the required level a few seconds after start. This overly rich mixture and retarded timing does create a rougher idle although in my Z06 it also creates a nice bark at startup with a rumbling exhaust so it has its benefits
Any engine that suddenly goes to a noticeably rougher idle has an issue. It could be an injector in need of cleaning, failing coil pack or plug, etc. and should be investigated but first I would try dosing a couple of tanks of fuel with Techron which does a good job of injector cleaning. This is the primary additive used by "Top Tier" fuel suppliers and although with direct injection cars it doesn't do anything for cleaning the intake valves it does a good job of cleaning the injection system and will also often bring a failing gas tank level transducer back into a reliable life.
Much of current emissions seems like backwards land with gas engines purposefully expelling unburned hydrocarbons to heat the cats and diesels injecting raw fuel into the exhaust stream every few hundred miles to incinerate collected soot in their particulate filters. I hate that with my new diesel pickup; cruising at 75 MPH the instant MPG stays in the 21-22 MPG range but when regen starts it immediately drops to 12 to 14 MPG when the 9th injector is used to dump fuel in after the exhaust side of the turbo. It only lasts about 20 minutes but it is annoying and wasteful. A DPF has a life of 150-250 thousand miles and then it is usually just scrapped and has to be treated as toxic waste, that is SO environmentally friendly