On a sphere, the lines of latitude and lines of longitude will always meet at right-angles.

I agree that they won't on a plane.

They do on both, obviously you are misunderstanding the FE model.

Draw 2 circles, do it.

Then from the center point, draw 2 lines radiating outwards, doesn't matter where

The angle where the line meets any circle will be 90

^{o}Every point on a circle is at 90

^{o} to a line from the center, that's how circles work.

What you have done is drawn a perpendicular bisector of a tangent.