You can quench copper and it doesn't really matter in my experience, but ordinarily you allow it to air cool.
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I had a rounded off diff drain plug on a van once that i couldn't get out,i used to siphon the oil out overnight.
From memory i think i used to get a piece of 1/4" copper tube bent around so it got to the bottom of the diff,then push a piece of clear tube over the copper and suck on it to get it started,then leave overnight,as it was bloody slow.
before you take them out, during which process they can be damaged, buy some serious duty brass plugs fully slotted from British Offroad.
They will post them to you straight away.