So of course both of these are slight simplifications, but what is the connection between the two? If the earth is basically a circle, is an ellipse just a parabola stretched around a circle? Is a parabola just an approximation of a tiny part of an ellipse? How high do you have to be before you change your calculations of a trajectory?
The Math ain’t mathing.
I don’t think that’s true. Any of it, really.
A parabola has a constant second derivative, while a (half) ellipse has one that diverges to infinity in finite time (since it goes vertical). I can’t really prove that no ellipse section is parabolic, off the top of my head, but that’s strongly suggestive of it.
Every two-body Newtonian orbit in a vacuum is stable, or an escape trajectory. It was either Newton himself or a contemporary that established that.
First paragraph: I see what you’re saying and I think you’re right.
Second paragraph: Even if you’re right on this, and offhand I’m really not convinced you are because it seems like that neglects the possibility of collision, launching anything from the surface of a planet is a three-or-more body problem, featuring 1) the planet, 2) the star it’s orbiting, 3) the launch body, and optionally a number of moons. But that’s, ah, getting away from OP’s question.
Well, exactly, in the context of OP’s question we’re doing “spherical cows in a vacuum”. In reality, no orbit is truly elliptical, either. Then again, if you use a practical measure for a practical situation, they’re close enough to elliptical for most space travel purposes, and definitely stable far away from an atmosphere.
General relativity can also cause noticeable departures even in our solar system, like the precession of Mercury’s orbit. In extreme situations it can get really different - gravitational waves remove energy, and around a black hole there’s a region where escape is possible but not any orbit.