Is it safe to short (discharge) an AC capacitor before you remove it from the circuit. Or do you have to wait until after you remove it from the unit?
Always short the capacitor as early into the disassembly process as you can. You may accidentally discharge it when handling it or removing it from the unit, and these components have enough energy to kill you.
I make sure to wear jeans and leather boots with a rubber sole when discharging capacitors, and always when it's relatively dry out. I keep my left hand behind my back and short the capacitor leads with a screwdriver that has an insulated handle. After about ten seconds of shorting, I double-check that the voltage across the leads is zero before considering it safe.
I wouldn't short the cap to discharge it. When you do that, a huge current flows for a very short time. This current is almost certainly far beyond the rated current capabilities of the cap. You may not destroy the cap, but you are overstressing it and shortening its life.
I would recommend disconnecting one terminal of the cap first, then shorting it through a resistor applied to both terminals. The resistor limits the current flow, while disconnecting one side prevents you from accidentally shorting/improperly loading your power supply if its still hot or turns on without warning.
In most cases properly designed electrical equipment will have built-in provision for draining the capacitors. So you shouldn't need to drain the capacitors, only verify they are drained, which you can do with your voltmeter.
If you do need to drain a capacitor, then best practice would be to drain it through a high value resistor.
In my opinion the least bad set-up would be two test probes connected via suitable resistors. A chain of ten 2K 0.6W resistors seems like a reasonable choice (chains of resistors have the advantage over single resistors that if one fails it's not a disaster)
Be aware that electrolytic capacitors can partially re-charge themselves if they are left open-circuit after discharging. Most capacitors used directly on AC won't be electrolytics though, electrolytics are normally found in DC applications.
Unless the capacitor has some kind of connector on it I don't think trying to remove it from the circuit while still charged is a good idea. Too much risk of an accidental short.