There is no significant fission reactor in the Earth's core. If there were, we could detect the antineutrinos from fission product decay.
I'm not sure where this idea came from. Perhaps from the mistaken notion that uranium, being heavy, sinks into the core? Uranium is actually highly enriched in the Earth's continental crust, by a factor of about 1000 vs. the planet as a whole.
Yes, and considering the cost of a nuclear plant a mile beneath the earth, just spending on geothermal would probably be cheaper, but "go deeper" isn't quite so simple either. Even in places with above average volcanism or a thinner barrier between the surface and mantle, you'd need to drill at least a couple miles down unless you're very lucky, and that extra mile (as a likely minimum) makes a big difference on cost and effort.
The plasticity, or ductility, of rock increases at higher pressures and temperatures, making it harder to drill through it, thus making it rapidly more expensive for additional units of depth beyond a certain point.
