Do you go under for eyelid surgery, or do they just numb your eyes?

Either. Going under anaesthesia is more expensive but more pleasant.
Upper or Lower? I perform all of my upper eyelid blepharoplasty procedures under a local anesthesia in the comfort of my office. Lower lid procedures (particularly ones performed from the inside of the eyelid) are best performed with some more sedation (iv or general). That being said, my partner does do some lower blephs under local in the office.
Anesthetic options. Blepharoplasty can be performed with local anesthetic alone (numbing shots), or with the addition of oral or IV medications to help with anxiety and pain. If other procedures are performed, general anesthetic is an option. Your surgeon can tell you what he/she usually recommends. Best wishes.
Either. Most lid surgeries can be done under local, some patients prefer to be put to sleep for it.
Up:yes/ lower:rarely. For upper eyelids, local is commonly used alone and/or with sedation. Some surgeons prefer local with ptosis repair. Lower eyelids tend to be more invasive and therefore sedation or general is more commonly used.
Local usually. This is typically done under local anesthesia but often augmented with sedation.
Usually local. Eyelid surgery can be performed under strict local anesthesia. General anesthesia is usually reserved for children and mentally handicap patient due to their lack of cooperation.
Depends. It depends on your surgeon and the procedure that you are having.