Are there different types of ileostomy surgery?

Yes. There is primarily one type of ileostomy commonly performed, the straight or brooke ileostomy. I am not sure if many are performing a koch pouch ileostomy anymore, which is a type of pouch used under the skin.
Ileostomy. Yes. A loop ileostomy will have two openings so it can decompress the colon and it can be put back together easily. An end ileostomy only has one hole and is typically done when the colon is removed.
Yes. There are a number of them done in the us and proponents for all of them.