When you get surgery for a cyst does that mean your whole ovary doesn't work anymore or its removed?

Depends on cyst. Most ovarian cysts can be removed or opened and drained with laparoscopic surgery. In rare cases the entire ovary is removed. Talk to your doctor about the options before your surgery.
Depends. If just the cyst is removed from the ovary, the ovary will continue to function normally. If the ovary is taken out with the cyst, obviously no more function from that side, but the other ovary will often compensate for the missing one.