Do they routinely check for STDs before doing cosmetic surgery?

Depends. It depends on the surgeon and you personal medical and social history.
No. In general, most cosmetic surgeons do not routinely check for std's prior to surgery. However, if a patient has a history of stds, especially herpes, many surgeons will place you on medication to prevent a flair-up. If you are concerned then i would discuss your concerns with your board certified plastic surgeon.
Not typically. For routine cosmetic surgery, not involving the genitalia, routine testing does not generally include evaluating for stds.
Yes. All patients who undergo general or mac anesthesia in my care, have below knee compression stockings placed and pneumatic compression stockings placed on top. In addition, patients who will remain on bed rest for 3 or greater days after surgery or who have a clotting history, or on going cancer or significant peripheral veinous disease also receive anticoagulant therapy.