Please explain if it is possible to have your teeth surgically moved?

Yes, it can be done. A tooth can be luxated (surgically moved or positioned) as an adjunct in orthodontic treatment. Movement can also be achieved by exposing an impacted tooth and placement of a bracket with a chain on the tooth to allow for orthodontic movement. Thin cuts in the bone between the teeth can shorten the time interval for orthodontic treatment. A tooth can also be transplanted in certain situations.
If. If you're talking about major full mouth tooth movement, where each tooth is cut out of the bone, repositioned, and fixated in place, not wise. Surgery as an adjunct to orthodontic care is often both prudent and advisable. Discuss your case with both an oral surgeon and an orthodontist.
Yes, it is. Oral surgeons are specifically trained and educated to remove teeth surgically.