Is it common for hospitals to give epidurals or general anesthesia for knee surgery?

Knee surgery: The decision is not up to the hospital - they have no roles in this. It is purely up to the anesthesiologist and and the surgeons preference. Discuss with the anesthesiologist first.

Depends : This decision is to be made between you, your surgeon and your anesthesiologist. The risks and benefits of your anesthetic plan will be discussed. Epidurals and general anesthesia are relatively common practice in the setting of total knee replacements. Femoral nerve blocks and catheters are also another option.

Both are common: It would depend on the needs and expectations of the patient as well as the preference of the anesthesiologist and surgeon.

Yes: The regional anesthetic has a protective effect on the formation of blood clots after surgery. There is also the ability to immediately exercise the knee though a range of motion while still getting pain medication. This allows an improved outcome after surgery.