I'm willing to bet that the portion of the exhaust that broke when you went over the pothole was already rotted. That could account for the loud shudder when you start it in cold weather (although the oxygen sensor loop is bypassed when the engine is cold, anything is possible) as well as the really rough sound. And, if the hole was opening up when you turned and affecting the signal being sent by the oxygen sensor it could easily account for the shudder when turning.
Fix the known problem. Get a good tuneup if you're overdue. Post back if you still have the problem.
Oh, and a broken exhaust system is not cosmetic. In addition to affecting your engine's operation, an opened up exhaust pipe can let carbon monoxide into your passenger cabin. That can put you to sleep without your even knowing it. It can kill you with a crash or it can kill you directly by displacing the oxygen in your blood. You need to get this fixed ASAP, before you get killed.