First, Welcome. Do a search on Backup Battery or Inverters, many of us have discussed this many, many times.
First, A stoker will not run at all without electricity. You need the combustion blower and stoker motors working to keep the fire burning.
Very Brief outages (less than 15mins or so), Yes, it MAY re-light, you may have enough heat in the bottom of the coal to re-light once the power comes back on. If you have a direct vent or power vent, it will probably go out much faster. A natural draft chimney may draw a bit more thru the stove and keep it burning a bit longer. Most stokers will shutdown after about 30+ mins or so.
YES, the stove will just push the coal into the ash bin if the power comes back on and you have a Flame-out condition. (Been there!).
A POWER Inverter or backup power supply can suppliment the burn time when power goes out. Most will run 30mins-3+ hours (Depending on batteries and power requirements of the stove). Most people use them to suppliment until they can get a generator online or other source of power. A TRUE SINE Wave Inverter are available from many different sources online, $100-$1200. Then you need at least one good deep cycle battery ($50-200), the more batteries, the longer the time you can run.
Various threads...Is there a 10 hour battery backup for stoker stove?Inverter for Homemade Battery BackupInverter Suggestions in Use Need Help