Yes, we left it burning. Had a log properly burning, made sure the door is on latch, and then we left it to burn out. Personal choice though - it would be safer not to, but when very cold we'll do this. It is however (even if you don't go to sleep if alight) essential that you have a working carbon monoxide alarm. We also have a fire extinguisher, fire bucket, knife etc in tent. Hopefully we'd never need to rely on any of those, but I think it would be irresponsible without.
As soon as the log is properly alight, it generates a lot of heat. I think to be safe I'd say 30 mins, max. What we do is to light the fire a short while before we go back into tent, as we keep the tent open for ventilation whilst starting the fire. It's not dangerous lighting it whilst closed, but I don't like the original smoke if everything closed up. Once the fire is properly alight you don't have the smoke coming out anymore. This might however just be my rubbish tent stove lighting skills, as it was the same when we lit fires in yurts.
We haven't used it for cooking in the tent yet, but we will make hot chocolate for early morning lie-ins and cook marshmallows over it next time. Again, we haven't used it for that, but you can remove the round stove plate thing and fit your kelly kettle over it to boil water, apparently. I love making a fire in kettle outside, but it would be nice if you want coffee inside or want to refill hot water bottles inside whilst you've got the fire burning anyway. Before we installed it in the tent, we did however make toast etc on it outside - we used it outside until I had enough courage to cut the actual hole.
Here is a review with a lot of pictures irl.