Is their a Hierarchy of Constructive Geometry?

In constructive geometry, there is a tradition that the purest 
constructions are those constructible with compass and straight edge 
alone, specifically a compass that collapses when lifted and an unmarked 
straight edge.

Under these rules, there are 3 famous problems that are known to be 
impossible:

Angle Trisection

Doubling the cube, which requires taking a cube root,, which I 
understand is somewhat analogous to angle trisection.

Squaring the circle, which requires constructing a transcendental 
length.

And there is the concept of constructible polygons, regular n-gons that 
can be constructed with compass and straight edge alone.

However, compass and straight edge aren't the only tools in the more 
open minded constructive geometer's toolkit. Some traditional examples 
include an angle trisector and neusis constructions, the latter which 
involves the use of a marked ruler and sliding it over a surface.

Neusis can be used to trisect arbitrary angles, but an angle trisector 
on it's own can't do everything a neusis construction can, so in a 
sense, an angle trisector is strictly weaker than neusis.

But are their tools strictly stronger than compass and straight edge and 
strictly weaker than angle trisector or strictly stronger than angle 
trisector, but strictly weaker than neusis? Are their tools that can do 
some of what neusis can that angle trisector can't while not being able 
to do everything angle trisector can?

Where do tools for constructing ellipses, parabolas, and hyperbolas fit 
into the hierarchy? What about tools for generating hypotrochoids and 
epitrochoids?

Is Origami greater than, less than, or equal to neusis?

A regular Heptagon is the smallest regular n-gon not constructible with 
compass and straightedge, but is constructible with angle trisector.

A regular hendecagon is the smallest regular n-gon not constructible 
with angle trisector, but is constructible with neusis.

A regular 23-gon is the smallest regular n-gon not constructible with 
neusis.

What is the simplest tool that can construct a regular 23-gon and what 
is the smallest regular n-gon this tool can't construct?