I am trying to create an application that needs to exchange small size of data - just to notify that the devices are close enough - for as long as the smart phone is running. I
Your question:
"Q0 Is it possible to have BLE turned on as long as the phone is running?"
is going to be platform specific. In Android I am sure that you can arrange to keep BLE scanning going on in the background, although that may be a battery concern.
On iOS the background application rules are more restrictive. I do not believe you will be able to continue scanning in the background at all.