Can you be "under anesthesia" for a gum scraping?

You can. But it will be more expensive that just having local anesthetic. Discuss your desires with your dentist and i'm confident you can be accommodated.
Yes but. It will be more costly, more inconvenient, has slightly greater risks, and unnecessary for almost everyone. However, it can be done. I would recommend "sweet air" in addition to local anesthesia and\or or oral sedation be tried first.
Possibly. Talk with your dentist to see if they have general anesthesia or sedation that might make your dental visit more comfortable.