No, it can not be an actual volcano, active, dormant, or extinct. Volcanos require HUGE amounts of heat and pressure from below.
I absolutely agree. I have thought of this myself.
[what i was always interested in is how different landscape elements formed.
As some others have mentioned, I like to think of this formation as more of crater than a volcano. Depending on just how high up the 'ceiling' is, perhaps a chunk of rock broke off, fell, and impacted the 'ground' (similar to a meteorite impact) to create a crater. To me anyway, it makes much more sense than a volcano.
As far as a name goes, I usually just call it 'the crater', but 'Crater Mountain' seems fine too. 'Official' names for some other landmarks would be nice as well. For example: the stone face (AKA. screaming rock, AKA. the stone mouth, AKA rock face, etc., etc.).