Short answers, pulled from the story.
The sphere of influence (SOI) in astrodynamics is the oblate spheroid-shaped region around a celestial body where that body exerts the dominant gravitational pull on a passing object. It is used most often to describe the zones in the Solar System where planets govern the orbits of nearby objects despite the far greater but more distant mass of the Sun.
Neptune's primary sphere of influence, roughly 53.5 million kilometres in radius, exceeds Jupiter's roughly 48.2 million kilometre radius because the SOI depends on distance from the Sun as well as planetary mass. Jupiter's much closer proximity to the Sun compresses its SOI boundary, even though Jupiter is far more massive than Neptune.
In the patched conic approximation, a spacecraft's trajectory is split into segments at each SOI boundary. Inside a planet's SOI, that planet is treated as the only gravitational influence; once the craft crosses the boundary, the Sun takes over. This converts the complex multi-body problem into a series of simpler two-body calculations using ellipses and hyperbolae.
No. The sphere of influence is technically an oblate spheroid rather than a perfect sphere. The distance to the boundary varies with the angular direction from the planet, and a more precise formula accounts for this directional dependence. The spherical approximation holds when one mass is much larger than the other.
Mercury's deep position inside the Sun's gravitational well produces an anomalistic precession of its perihelion that Newtonian gravity cannot fully explain. Albert Einstein's general theory of relativity, which ties gravity to spacetime curvature and the speed of light, correctly predicted the precession. Mercury's orbit became one of the first observational tests confirming the theory.
The sphere of activity extends well beyond the sphere of influence and should not be confused with it. The SOI marks the region where a planet is the primary gravitational influence on a passing object; the sphere of activity is a related but larger boundary used in other orbital mechanics contexts.