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

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.