existing name for name for a "castable" 3d object? For our purposes, consider an object castable (as in 'cast in a mold') if there exists a plane such that every line orthogonal to that plane passes through the surface of the object exactly zero times or exactly twice.
We can also have a variation: A plane exists such that every ray orthogonal to this plane  originating from this plane passes through the surface of the object exactly 0 times or exactly once. This is stricter because it eliminates certain types of overhangs, and is equivalent to saying that there exists a plane dividing the object such every point of the object's surface projects onto this plane without that projection ray passing through any other part of the surface.
For example: A solid sphere is castable, for instance by bisecting it. A torus is also castable by both definitions (the plane cuts it like you cut a bagel). A partially hollowed sphere (like a soccer ball - remember it's filled with air and has a thickness) with an inner and outer surface, is NOT castable by either definition, since some line perpendicular to any plane will pass through the surface 4 times. If you cut this hollowed sphere in half then that half is castable by definition 1, but not by definition 2.
 A: This sounds like some kind of convexity condition. One way to define convex sets is: A set is convex iff every intersection with a line is connected. This gives rise to all sorts of generalisations.
For example star convexity. Here we only look at lines trough a certain point, but the rest remains the same: A set is star convex iff every intersection with a line trough a fixed point $p$ is connected.
Another type of convexity is ortho-convexity. This is very close to what you are talking about. A set is ortho-convex iff every intersection with a line that is parallel to one of the coordinate axis is connected.
In Restricted-Orientation Convexity by Eugene Fink and Derick Wood, this is further generalised to $\mathcal{O}$-convexity. The lines with which we will consider the intersections are the translations of lines in some set $\mathcal{O}$. If you choose $\mathcal{O}$ to be the set of the 3 coordinate axis, then you get ordinary ortho-convexity. 
If you chose $\mathcal{O}$ to be a single line orthogonal to the plane, then a set is $\mathcal{O}$-convex iff and only iff every intersecion with a line orthogonal to the plane is connected.
If I understood your condition correctly, then a castable object is an object such that its union with a plane is $\mathcal{O}$-convex, where $\mathcal{O}$ is the set containing one line orthogonal to the plane.
