As previous posts, if battery is being dragged down by starter, it may not be cranking quick enough for the ECU to allow the injectors/pump etc to operate. (they have a minimum cranking speed, unless this is reached the car won't run)

Screech could just be the auxilary belt because of the 'rough' cranking/shaking.

It's always the first start of the day when the oil is stone cold etc when the battery/starter gets the most load against it.