I just tried my oven after seeing your question. I turned both oven and a stove top burner on then unplugged it, the oven shuts off, and the burner remained lit.( I assumed your question referred to an outage while using oven.) However if there is an outage and you want use stove top, you would need to manually lite the burners, and oven wont work at all.
Outside oven gets warm to touch when using, not too hot. Not sure of propane conversion kit, however do think there are common...