No. That's local anesthesia. It's called a "nerve block". The anesthetic is injected right next to a nerve. If you imagine pain as starting somewhere, then, "travelling" upstream to your brain, the anesthesia basically interrupts signals from the point where the injection takes place.

Awesome, thanks for the explanation, that makes me feel better about going to the dentist.

I think local anesthesia does not lead to these effects (as far as we know), I also would be surprised if it does. One thing I remember when you also use N20 (laughing gas) you should either check or just take vitamin B supplements (it seems to deplete your vitamin B storages)

No, that's local, not general anesthesia.