If the head gasket is leaking it is possible that water get into a cylinder.

The most common symptoms are:
1- very hard first crank after the engine was not running for a while (overnight).
2- Large amounts of white smoke from exhaust
3- Water in the oil (oil level rising)

If your car is just idling bad for 20 sec I would recommend further diagnosis before you have the engine taken apart.

You state that the coolant indicator comes on sometimes, do you add coolant or just wait for it to go off again?