I30 Stalling and Acceleration Problem

What is your question?

We have a 2000 Infinity I30. It is idling erratically and occasionally stalling when the car is not in motion. Also if you punch the accelerator and have it downshift gears to rapidly gain speed the engine bog's down and will not let you accelerate. You can accelerate but it has to be done slowly. We have read that it would be all kinds of things. We were wondering since we have the stalling problem and the acceleration problem if those two are linked and if there is a common denominator that would be aiding in the problems. The check engine light comes on and goes off periodically. The problem is consistent whether the engine light is off or on. We did get a reading and the code was P0134. It gave us an oxygen sensor fault. The the problem that i am having is if it was an O2 sensor wouldn't we be getting horrible gas mileage? The gas mileage hasn't changed at all since we have had the problem. Any thoughts?