If I remember correctly - and it's a very long time since I had anything to do with it - On Train PA announcements can be made via GSMR. So those equipped with the right equipment in signalling centres or the like could make announcements directly to the train, assuming there is still enough power for the GSMR and PA systems.
Yes they can. I've been on a train where a driver has left the cab to deal with something and the signaller has been able to request the driver returns to the cab.
Maintaining the GSM-R, PA system and some basic lighting is a must. If a train could be stranded for 5 hours or more in extreme cases, the battery should be able to cope with that, or even longer. It really wouldn't need to be a significantly large capacity battery in the grand scheme of things (I am not suggesting it would be used for traction). A 500Wh or 1kWh battery should be ample to power such systems for ages. Obviously this depends on the length of the train and how many lights you'd need, and a 9-12 car fixed formation train would have a bigger battery than a 4 car - with 8 or 12 cars made up of 3 trains each having their own battery (and so on).