Yes. Laparoscopic surgery generally always requires general anesthesia due to the insufflation of the abdomen with co2 to create a space to see and work. Removal of an iud does not usually require laparoscopy though, unless it has somehow migrated intraperitoneal. Iud removal can often be done with a paracervical block, with or without sedation.
Yes. General anesthesia is recommended for this hopefully short procedure. Sometimes it takes some time if the iud is free in the abdomen & I've had some interesting cases that required meticulous inspection before finally finding it hidden in bowel or omentum or even snuggled up by the liver. So, yes-have general anesthesia.
Yes. Laparoscopic abdominal surgery requires general anesthesia.
Yes. Because laparoscopy involves filling your abdomen up with carbon dioxide gas, you will need ga to help breathe against the pressure. Once the procedure is done and the pressure removed, the breathing tube will be removed.
No. This can be done with local, with a small amount of sedation. Only if the iud is really stuck would they have to go deeper.