Prove that only fields involving square roots are possible with ruler and compass and find fields that are possible with other conic sections.

Question

Around the 16:40 mark of [1], Mathologer tries to convince us that only "square-rooty" fields can be achieved with ruler and compass. He first shows that any field which involves combining rational numbers with any amount of square rooting of rational numbers (nested or in combination) is possible with ruler and compass. So far so good. He then argues that this is the limit of the method. The argument he uses for this (almost certainly over-simplified for the sake of the video): "any line through two points with square-rooty coordinates has square-rooty coefficients and any circle with center at a square-rooty coordinate and passing through another square-rooty coordinate has square-rooty radius. And the points of intersection of these lines and the circles are linear and quadratic equations, their solutions are also square-rooty". This put a bit of a simplified picture in my mind which was shattered when I came across this post: Doubling the cube with the help of a parabola. I had assumed that parabolas were still quadratic equations. So, the argument around systems of quadratic equations producing square-rooty fields should still hold. Yet there is a way to cube root 2 with them (and not with circles).

This opens up some other natural questions:

1. How do we prove that this can never happen with circles? I understand vaguely that the parabola can be quadratic in $y$ but linear in $x$ and this is what makes the cube possible. But how do we make it rigorous?
2. What is the field we can achieve with lines, parabolas and circles.
3. What about fields we can achieve with lines combined with the other conic sections one at a time (I assume ellipses will generate the same field as circles since they are just scaled circles, hyperbolas are interesting since they can have $xy$ terms - does this give them the same field expanding powers as parabolas)?
4. What is the field we can achieve with lines combined with all conic sections (which means all quadratic equations)? My guess would be fields constructed with rational numbers and any root of rational numbers that are factors of 2 or 3 or combinations thereof. But how do we prove this/ show otherwise?

[1] https://www.youtube.com/watch?v=O1sPvUr0YC0