Why does the car stop running after idling for about a minute?

If you are able to retart easily and keep the engine runing with a little throttle pressure, it means the idling speed is too low and usually is due to partial clogging of the idle conteol system air passages. Getting the throttle body and IAC cleaned should solve the problem.

If the shaking is coupled with poor performances, then it could be a misfiring engine and you need to check the spark plugs and related components.

If the MIL is indicating, then you need to retrieve the trouble code to see what is affecting the engine.