Raise your front end one wheel at a time, and check for any play in the wheel by shaking it side to side, and top to bottom. If you have any play what so ever, then start looking at your lower ball joints, from there you can check your bearings. The way I diagnose these things is in the following order:
1)high speed wheel balancing -did they take off the old weights from the rim, before they re-balanced the tires;
2)shocks - check tires for steps on thread , if tires feel round, press on the shock, it shouldnt bounce more than two times, if it does, its becomming, or is loose, if you have steps in your tire, your shocks are badly worn out.
3)lower ball joints - check for any movement of the whole wheel. Any slack there would point to lower ball joints, or possibly bearings, although bad bearings usually do not cause slack, but rather a whinning noise.
4)damper - remove the damper from one side, and see if you can move the arm, if it moves rather freely, it needs replacement
5)wheel alignment - normally, I get a wheel alignement when I replace the tires, make sure your tire pressure is up to spec, before you go get an alignment

I am not sure how the offset of that one tire is affecting your car's performance, and transmission, but I would replace that rim, to match the other 3. If you get to #5, and the issue is still not resolved, the person doing the wheel alignment should be able to diagnose the issue for you on the spot. Hope this helps