When I plug in a dimmable lamp on the other side of the house, a GFCI outlet in my kitchen starts humming. What gives?
Since the GFI's test button does not make it trip, the GFI should be replaced pronto.  Until that can be done, unplug everything from it and from downstream outlets.

The test button is not an accessory feature.  It uses simple electronics to produce a slightly unbalanced current so as to directly test the protection logic.  If that doesn't work, the protective circuitry has failed.  That, plus any buzzing (especially under bizarre circumstances) is a screaming cry to fix it.