I know Team Fortress 2 has a StudioMDL (at least for Windows), so if you have that installed, try that.

I don't know for the Source SDK Base 2013 SinglePony nor MultiPony, but I for sure know that Garry's Mod does not have a StudioMDL.exe. Could you link to the full path you set for the Source SDK Base 2013 SinglePony as Engine path?

No idea what the problem may be. (I have a 64-bit Windows 7, running Blender 2.69, and Source Tools 1.10.4, which works completely fine.) Maybe the author can figure it out the next time he logs in to Steam.
And I use the add-ons Blender has enabled by default, plus the Blender Source Tools 1.10.4.

I don't think you told us what operating system and such you were using. And other plug-ins.

I don't think you told us what version of the Blender Source Tools you used...

You CAN also use B (boundary selection) or A (select all) or something to select all bones, and move them all at the same time...

.smd requires every keyframe to be there, yes, however, I think Blender smoothens out missing keyframes. Plus, .dmx doesn't require a keyframe in every frame. And .dmx fills less.

You know, you CAN also just select all bones at once in Pose Mode, and then move them.
Unless you have vertices not weight-mapped completely, that would do the same as object mode translations, except that the Blender Source Tools would accept that.
(However, transferring the object mode translation to pose mode... no idea about that.)

As the other exported animations work fine, while not using that movement technique thingie, a work-around would be to not use that technique, and only use pose mode keyframes. (The Blender Source Tools completely ignores object translating/rotating, and only thinks of the edit mode mesh, and pose mode position/rotation.)

I guess so, too.