It's been over 15 years since I learned to degauss a TV (we were still fixing vacuum tube TVs at the time). What I was taught was significantly different and may not apply to modern monitors. We degaussed using circular motions along the top, bottom, and sides, never on the face.
As far as when to turn the coil on and off, I think it's probably not critical when you turn it on (since the "on" EMF pulse is minimal and you're going to degauss after turning it on anyway), but you absolutely must keep it turned on until you back away from the monitor because turning off the coil causes a very strong magnetic pulse which will ruin your degauss if you're too close.