Is there some simple description of which complex numbers are "constructible" with straightedge and compass and neusis?
See http://en.wikipedia.org/wiki/Constructible_number and http://en.wikipedia.org/wiki/Neusis.
[Math] Neusis constructions
ac.commutative-algebraeuclidean-geometrygeometric-constructionsmg.metric-geometry
Related Solutions
Have you found the papers by Roger Alperin? He has some very nice articles, especially in regards to relating various construction systems -- in addition to origami and compass-straightedge constructions, there are a variety of others, like the Vieten constructions, Pythagorean constructions, and even some variants on the origami constructions. Basically, each additional axiom potentially provides an entirely new class of constructions (which may or may not actually enlarge the set of valid constructions). Some of these questions end up being very geometric in nature, and others have some fantastic ties to the structure of algebraic numbers.
In any case, you can find several papers of his via Google. I believe your specific question about intersecting two circles is answered in Section 6 of Alperin's "Mathematical Origami: Another View of Alhazen's Optical Problem."
Hope that helps.
A bounded-length straightedge can emulate an arbitrarily large straightedge (even without requiring any compass), so the rusty compass theorem is sufficient.
Note that, in particular, it suffices to show that there exists an $\varepsilon > 0$ such that a straightedge of length $1$ is capable of joining two points separated by any distance $\leq 1 + \varepsilon$ (and therefore emulates a straightedge of length $1 + \varepsilon$, and therefore arbitrarily long straightedges by iterating this process).
We can use Pappus's theorem to establish this result for any $\varepsilon < 1$:
https://en.wikipedia.org/wiki/Pappus%27s_hexagon_theorem
In particular, given two points $A$ and $c$ (separated by a distance slightly greater than 1) which we wish to join, draw a line $g$ through $A$ and a line $h$ through $c$ which approach relatively close to each other. Then add arbitrary points $B, C$ on $g$ and $a, b$ on $h$ such that the four new points are within distance $1$ of each other and the two original points. We assume wlog $b$ is between $a, c$ and $B$ is between $A, C$.
Then we can construct $X$ by intersecting the short (length $< 1$) lines $Ab, Ba$, and construct $Z$ by intersecting the short lines $Bc, Cb$. Then $Y$ can be constructed by intersecting the short lines $XZ$ and $Ca$.
Now, $Y$ is positioned collinear with $A$ and $c$ and between them, so we can use it as a 'stepping stone' to draw a straight line between $A$ and $c$.
The result follows.
EDIT: I decided to do this with the edge of a coaster and two points slightly too far apart to be joined directly:
Best Answer
Just as straightedge and compass constructions give the numbers in the closure of the rationals under square roots, neusis gives the closure of the rationals under square roots and cube roots.
For more details, also for an alternate characterization in terms of origami, see this paper by Roger Alperin.