This question is a partial duplicate of this one and this one.
When ATC gives a heading, receives the correct readback, but observes a 10- to 15-degree difference between the actual and intended tracks, does the specialist incorporate this difference into future vectors?
Sometimes. The controller issues headings but observes ground track. If the wind is causing drift, the controller will compensate for it when she provides vectors. Sometimes, if an aircraft is suitably equipped, a controller will ask for a wind report in order to better judge the winds at altitude.
Do approach and traffic separation vectors generally have a healthy margins of error?
Yep. See here.
How does apparent compass deviation or directional gyro precession factor into vectors that ATC issues?
It's up to the pilot, not the controller, to correct for compass deviation (since only the pilot can see the compass correction card inside the aircraft) and gyro precession (since only the pilot can turn the little knob to reset the gyro).