Is there a way to subdivide a single edge between 2 vertices on a mesh without Blender adding several more edges to vertices I haven’t selected?
It’s annoying to say the least when all these extra edges pop up :x
I don´t think so. You could just remove the new edges. But imagine you will move the new vertex then, it might produce awkward results.
Depending on what you want to do, there might be better ways, e.g. extrusion…
since blender doesn’t support ngons (5 or more sided faces) the only answer is to add those edges, unless there are no faces there to begin with.
Select the edge, Shft-D to duplicate, W - subdivide, all without moving the mouse. Tab out of edit, Tab to select the whole mesh, A to select all, W remove doubles.
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This is extremely clever 
Unfortunately it doesn’t subdivide anything (move your newborn vertex to see that it is not attached to anything and that the face still is absolutely the same) yet it may be useful in some circumstances.
Jean
Yes, you’re right. The original edge needs to be deleted and in some cases a new face created. But i use it often, eg subdividing a loop. I just find it easier to delete the original loop and add the new faces than to mess with all those edges individually.
Will I ever be able to give some usefull advice where you or Theeth don’t have to clean up after me? 
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eheheh
Just sees that as double checking, you learn something new and the original poster of the question gets more answers than you can shake a stick at.
Martin
PS: Just thought I might also add that a bunch of clever people (cough) are working on edge selection for Blender 2.28 (although it might not be ready for the release).
/me crosses fingers
finally some real tools!