I'm not a geologist, but for comparison, the Siberian Traps[1] erupted for 2 million years, and there have been other large events on that scale. Any heat we release will be insignificant in comparison.

1. https://en.wikipedia.org/wiki/Siberian_Traps

Technically it would, but the same way an individual ant's breath contribute to global warming.

Note that the thermal energy contained in the inner earth isn't a fixed amount: the earth constantly generates new heat from friction and radioactivity, and that new heat is radiated away in space at night, alone the heat captured from the sun during the day. So technically geothermal will increase the efficiency of this heat transfer a tiny bit, but it's really a negligible amount. (In reality the ant's contribution to global warming is probably orders of magnitude than the phenomenon we're talking about here).

Not the core measurably, but my understanding is that it can very slowly cool the volume of earth immediately around the heat extraction site.

I have full faith in modern engineers' ability to account for these effects in the plant designs.

It could prevent volcanoes from erupting perhaps.

Nope. It is very far away from core. Core is very big. And energy is already flowing outwards. So just using it really have not any significant impact.

Real issue is that it is not actually usable everywhere. Again Earth is big and going deep in some places is very hard.