During cold starts all cars are programed to idle at a little higher RPM and once warmed up to settle to the normal range.
In your case there can be two issues one is the weather and the other fueling. The latter can be sue to the bad quality of the recently filled petrol so some block(minimal) to the injectors.(sprakplugs/timing comes next in line).
Continous vibrations from the engine can be due to faulty foundation bushes(engine mounts).
In your case the problem resolves once the car is warmed up-hence the possibility of the rich mixture(more petrol) at cold starts being the cause is high.
How many KMs has the car run till now?Has the air filter been cleaned/replaced at a recent service?
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