Can a standard battery be used as a backup power supply?
The problem is that initially the board will be supplied with power continuously.
Will this cause problems with the battery itself, or is revision D contain a charge controller to ensure proper battery handling?
It is very common to have a Li-Po battery connected to the board all the time! It is not a problem. It has a very good power management unit that handles the supplies. If you wish to know more about the power management unit search for "AXP803".
One thing to consider is that you need to check the + and - of the battery before connecting it to the board.
Here you can see a typical rechargeable battery compatible with the board and the charger:
https://www.olimex.com/Products/Power/BATTERY-LIPO1400mAh/
Or if you expect longer lack of main power supply:
https://www.olimex.com/Products/Power/BATTERY-LIPO4400mAh/