No. Most facial operations do leave a scar, although they are usually hidden within creases or wrinkles. However, some operations do not leave a scar, such as laser skin surgery or arthroscopy. Some facial operations are completely hidden by being done through the mouth or nose.
Yes. The goal of the surgeon is to leave them in inconspicuous areas and to try to make them as short and narrow as possible, with no suture marks (dots on either side, crosshatching). Learning to do this is a large part of the long training of a plastic surgeon. All surgeries, though, leave scars.