Consider a large set of points with coordinates that are uniformly distributed within a unit-length segment. Consider a Voronoi diagram built on these points. If we consider only non-infinite cells, what would be (if any) the typical (most frequent) number of edges (that is, neighbors) for those cells? Is there a limit for this number when number of points goes to infinity?

Consider the dual of the Voronoi diagram, which is the Delaunay triangulation of the given points. This is a planar connected graph, so its Euler characteristic is $\mathrm{\chi <!--\; \chi \; -->}=2$. The Euler characteristic is also given by $\mathrm{\chi <!--\; \chi \; -->}=V-<!--\; -\; -->E+F$, where $V$, $E$, and $F$ are the number of vertices, edges, and faces in the Delaunay triangulation respectively. Now every face has $3$ edges, while all non-boundary edges are adjacent to $2$ faces. Under reasonable conditions*, the proportion of boundary edges tends to zero, so let us ignore them. This means that $3F\approx <!--\; \approx \; -->2E$, and plugging this into $V-<!--\; -\; -->E+F=2$ gives $V\approx <!--\; \approx \; -->\frac{1}{3}E+2$, where by "$\approx <!--\; \approx \; -->$" I mean the ratio tends to $1$ as $V\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}$. So there are about three times as many edges as vertices, and since each edge is incident on $2$ vertices, the average degree of the vertices approaches $6$. As the degree of a vertex in the Delaunay triangulation is precisely the number of edges of the corresponding Voronoi cell, this agrees with your intuition for the $2$D case.
In three dimensions, the Euler characteristic is still $2$, but is now given by $V-<!--\; -\; -->E+F-<!--\; -\; -->C$, where $C$ is the number of tetrahedral cells in the triangulation. A similar argument as above gives $4C\approx <!--\; \approx \; -->2F$, but we have no control over $E$. Indeed, there are triangulations on the same point set with the same boundary (the convex hull) but which have different numbers of edges. In the $2$D case, every such triangulation had exactly the same number of edges.