How far can A&P mechanics operate aircraft under US regulations? Anytime an A&amp;P mechanic wants to test the updates/fixes made to an aircraft's systems or any part for that matter, what are the laws (at least under the FAA) that legally allow them to operate an aircraft?

I've personally seen GA airplanes being taxied (in non-movement areas so far) by A&amp;P mechanics. Is there a law that states that they can/cannot operate that very aircraft on a taxiway (movement area)?

What about larger aircraft (business jets, airliners etc.)? Do authorized mechanics always need a licensed (and appropriately certified with the required ratings) pilot all the time when testing the aircraft after repairs?

Finally, what about military jets? I once acquainted with someone who worked on military jets' electrical systems, but was never licensed to fly. Would he solely rely on the PIC's conclusion about the condition of the jet?
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.