standard "engine fire on startup" procedure in thye POH of one of my aircraft is "continue cranking the engine" - this supposedly uses the excess fuel up.
On the ones I fly too. If the fire is somewhere in the intake system, then this sucks the fire into the engine, where it can't do any damage.
But the next items are: shut everything off and evacuate, if possible taking the fire extinguisher with you. Not: go flying as if nothing happened.
Sternone, if you were not happy with what happened (and the way I read your post, you weren't), you should not have accepted the aircraft and gone flying. That's what command authority is all about. Taking the cowling off and giving things a good look over will only take a few minutes and you already had the tech guy around.
As for taking off with a suspicious battery, well, it's not dangerous per se in one of those old spamcans. But it would be a nuisance if you had to shut the engine down somewhere (on a landaway, for instance) and would not be able to start it again. Then again, it might just have been a cold and slightly depleted, but otherwise perfectly healthy battery. You do keep your walkaround, with lights flashing, pitot heat burning etc. as fast as possible, don't you?