2001 Questions for all other cars stalling

I recently bought a Rover 45 petrol automatic. Its in good nick and appears well looked after. I do notice the tell tale signs of cylinder head overhaul from the silicone smear on the protruding gasket edge etc, however the previous owner tells me the engine was never touched. It runs well and generally have no problem, only when you start it in the morning it has a rough tone on the idling until you pick the revs up slightly. Then it disappears.( In PARK and cold). The main hassle is when its cold and you drive a bit and stop at a intersection say, it stalls easily. The problem disappears when its warm (5 minutes). Also, even when warm and you stop at an intersection, its prone to rocking or vibrating slightly.
This being my first automatic, I am not sure what to make of this.
Can you help. (Am sure you can)
Many many thanks