For fascinating read fantastic . WTF is an earth SOI . I'll assume that means "sphere of influence " .
So you can google, good boy.
Escape velocity according to Newtons laws is 11km/s . Anything less and your pulled back to earth . Anything more and your free of earth and off to wander in "space" .
Wrong, leaving Earth's SOI just keeps you in a solar orbit, not wandering around. Not reaching escape velocity does not assume you will fall back to earth unless you are on an extreme parabolic trajectory. In the case of a spacecraft, they are put into an Earth orbit and then the apogee is pushed out beyond the SOI.
Schoolboy stuff . Velocity is a vector quantity - magnitude and direction . Angular velocity requires a constant force . Feel free to direct me to an explanation of all said vectors contributing to orbital mechanics . The n-body scenario is a problem that requires all variables ( vectors ) of all bodies within that system to be known exactly at the instant when the orbit of moving bodies is calculated . Space flight does not work as we all are told . Science cannot find any rotation of earth , nasa uses geostationary flat earth co ordinate system for launches and flights - because it works . Because the earth is stationary .
No one is going to use a so-called flat-earth coordinate system since one does not exist. The Earth is far from stationary as it is orbiting the sun which orbits the galactic center which is moving away from the origin point. Prove it's not and we can continue.
So how , when the satellite escapes SOI does it then accelerate to 66,600mph ( about 30km/s ) to catch earth if it has escaped from SOI , in a vacuum?
Already covered in as you say, basic schoolboy science that you still have wrong. Why would one need to accelerate to escape velocity if it already escaped? It would already be moving at that velocity relative to the sun. It is matching the orbital speed of the earth that is moving around the sun. Simple stuff really.