How far can A&P mechanics operate aircraft under US regulations?
The laws for an A&amp;P are the same as they are for a pilot. To fly the aircraft you must cary the appropriate licenses and ratings required for the airframe. No special cases are made for mechanics even for return to service flights. 

However mechanics often fire up the engines and may run them up outside of the hangar to test things. As far as I know no certified pilot is required to start up and run aircraft engines.  

The part about taxiing is partially addressed here. I know that mechanics can get some kind of approval to taxi aircraft if they do not hold any licenses, when I flew out of KPNE I knew some of the mechanics on the field and they needed to get some training on radio operations (as it was a towered field) before being allowed to taxi aircraft.  

The last limiting factor, that may be a bit more ambiguous is the insurance regulations. For example say you were to own a little C172, your insurance may stipulate that only you (and perhaps an instructor) was allowed to fly the aircraft. In this case it would be unwise for your mechanic even if properly rated to fly the airplane as your insurance policy does not cover that.