What about an arbitrary polygonal planar linkage? The non - convex
linkage? The linkage with any number of bars? What happens when not
all of the vertices can move?

Well, in some cases (like the linkage in the figure), we can see that the linkage is unopenable. But then again, this is not a very interesting linkage, and you wouldn't want to use it as a mechanical device. What about the case for interesting, worthwhile linkages other than the four - bar linkage or the convex linkage?

The fact is, nobody knows. All the problems related to
straightening planar linkages are open problems. What classes of
linkages can always be opened? Other than convex and four - bar
linkages, nobody knows. Is there some arrangement of movable vertices
that allows any linkage with that pattern of vertices to be opened?
Nobody knows. Can every linkage with each vertex movable be opened?
Again, nobody knows. Are there any classes of linkages that can
*never* be opened from some position? Your guess is as good as
mine.

What about this one: Are there any linkages that cannot be straightened by moving one bar at a time, but that can be straightened if you move a group of bars at the same time? Tricky, yes. But a similar question has been answered for polyhedra.

What about if you want to open the linkage by moving the ends only? (In this case, any joints that are straightened are treated like they no longer exist -- the end is welded to its connecting edge.) Again, all the problems given above are open problems even with this restriction.

The figure gives a special case of a four - bar linkage that can't
be opened from the ends, but what does this mean in terms of different
kinds of linkages?

Why don't you try to find out? Play with our linkage applet -- make an arbitrary linkage, then move it. Be the first to solve this open problem in computational geometry!