Follow the directions your doctor's office gave you. You should perform range of motion exercises according to your doctor instructions. Range of motion should not injure your surgery, if your doctor prescribed it as a post-operative exercise. Generally we recommend cold for at least the first 6 weeks. Hot tubs may help with individual problems and can be recommended earlier, but only by a physician.
Surgical stockings serve at least two purposes.
They control edema and help prevent blood clots. They should be worn for 10-14 days following surgery. And yes, clots can form in the other leg, so wear them both. That's a tricky question. After surgery, there is often extra swelling in the knee that adds pressure behind or under the kneecap.