> The earth is pretty sphere-ish, locally speaking.

Kinda. There are mountains, they have their own gravitational attraction, and it can be measured... even in the 18th century! https://en.m.wikipedia.org/wiki/Schiehallion_experiment

It all depends on how pedantic one wants to be :)

And a ball thrown in the air follows a parabola, locally speaking. At least, you're better off correcting for air resistance before you sub in the ellipse equation.

Only if the sphere has uniform density. The Earth very much does not have uniform density.

Locally speaking, the earth is flat.