Experienced elbow pain the day after I played rough tug of war with dog. This happened a week ago, but elbow still hurts when I use it.

Elbows are easy to hurt, and they usually recover without medical help.
Get some OTC pain reliever like ibuprofin, some cream pain reliever like Icy Hot or Bengay, keep the elbow both rested but also moving gently frequently. When you sleep, try not to lie on it. Drape that arm over an extra big pillow. See if you can gently massage it while moving it. GENTLY.
If there is no redness or swelling, let it go another week.

The trouble with elbows for a doctor is that there are muscles, tendons, ligaments, nerves, blood vessels, and bone. Unless the bone chipped, nothing will show on X ray. So mostly you are the doctor. Use your good judgment after another week. Even a bone chip isn't operated on. You might get a sling. You can make or buy a sling. A tear of a muscle/tendon/ligament usually hurts the most, and again, that takes time. If it's a worse tear than usual, time will tell.