you can start with projective geometry which basically has just an ellipse.
In general I'd classify quadrics in projective geometry into several classes, too. See this answer for an overview. What you call “basically […] just an ellipse” I'd call a real non-degenerate conic, which is one of several classes. Cubics would have more classes.
First, I think that cubic curves, unlike conics, exist in topologically distinct forms even in projective geometry.
You are right. See this answer for a discussion of why a single representative is enough for quadrics; that idea would not apply to cubics.
A quadric has 5 real degrees of freedom. You can pick 6 coefficients or matrix entries or whatever, but scalar multiples describe the same quadric. A projective transformation has 8 real degrees of freedom, since it maps any 4 points in general position to any other 4, and each such point represents two degrees of freedom. A cubic has 9 degrees of freedom (10 coefficients to choose), so there has to be an infinite number of equivalence classes of cubics modulo projective transformations.
But once those forms are found, could I simply derive Euclidean cubics by making one line special and hyperbolic cubics by making one conic special?
Sure. A quadric or cubic is a first class citizen in projective geometry. The distinction into ellipse, parabola and hyperbola requires a bit of Euclidean geometry, namely the line at infinity. The classification of some ellipses as circles requires even more Euclidean geometry, but more on that in a moment. You could count intersections between a cubic an the fundamental conic, and then invent names based on that. Not sure anyone would be interested, unless you could come up with applications where those distinctions became relevant.
if Euclidean geometry can be obtained through special line and hyperbolic geometry through a special conic
Not quite. You need a conic in both cases. In Euclidean geometry, that conic is degenerate. As a set of points, it represents the line at infinity, with algebraic multiplicity two, i.e. the conic $z^2=0$. But that's just part of the picture. You can also describe a conic in terms of its tangents. Usually the matrix for the tangent description is the inverse of that for points, but in the case of degenerate conics inversion leads to division by zero, so it's better to represent this so called dual conic separately. For Euclidean geometry, the dual conic is $x^2+y^2=0$, which factors into the two circle points $[1:\pm i:0]$ (or more precisely the set of all the lines through these points) since $x^2+y^2=(x+iy)(x-iy)$. Every circle passes through these two points, and since circles are the basis for length comparisons, and you can also compute angles from these points using the Laguerre formula, the dual fundamental conic is crucial for metric aspects of Euclidean geometry.
The great unifying concept here is Cayley-Klein metrics. The idea is roughy as follows: to compute a distance between two points, draw the line connecting them, intersect it with the fundamental conic, compute the cross ratio of these four points. Dual to that, to compute an angle between two lines, find their point of intersection, draw tangents from that to the fundamental conic, compute the cross ratio of these four lines. In both cases you need to finish the computation by taking a logarithm and applying some constant and possibly complex factor, but that's just consmetic. And for Euclidean geometry all distances will turn out as zero, because that geometry has no absolute distance measure, just relative distance comparisons, but I'll not go into more detail.
Note that for other classes of conics you get other metrics. Richter-Gebert's Perspectives on Projective Geometry has a nice overview of the 7 possible situations here, plus more details on working with Cayley-Klein geometries for various applications. Elliptic geometry resulting from a non-degenerate fully complex conic like $x^2+y^2+z^2=0$ is one interesting class, the geometry of space-time relativity resulting from fully degenerate conics is another. Disclaimer: I've worked closely with the author.
is there a geometry that would be based on claiming a cubic as special
Based on the above, you'd need to replace the cross ratio of four points with something where you use five points as input: the two points on the line and the three points of intersection with the cubic. The formula would need to be sufficiently symmetric that changing the order of points does not change too much (although it does change the sign of the distance for e.g. hyperbolic geometry). Off the top of my head I can't think of a reasonable candidate to fit this bill. If you do find one, it might be interesting to see whether a definition based on this actually satisfies the axioms required for a metric.