Easy - Pro Tools.
Its standard practice to both gate or simply automate the track to mute at certain points.
You'd take a DI from the guitar for reamping in the studio. This way you can either gate/automate the amplifier without the effects (Steve apparently records dry, without effects) or modify the original DI signal. I regularly do both with great success.
As for live..
Believe me, there is a fair bit of noise. Its inevitable that white noise would be an issue, but its an easy fix; simply put a volume pedal in the loop. This stops any signal flowing to the speakers, eliminating the problem between songs or at dropout points in songs.
Buzzing and Hum can be fixed through ensuring everything is grounded and shielded properly (and a lot of other electrical hoohaa I won't bore you with!) and through quality equipment. Steve's FTLOG Universe has a fair bit of hum on the Stillness In Motion DVD, as you can hear at the start and end of Weeping China Doll.
Its all just experience and know-how with equipment