Heyo - I went through a majority of your bullet points, and they are all actually fine.
Be sure you're moving the right bones. These rigs (all the same btw), have both IK and FK rigs. Be sure you're showing the right ones by toggling on the bone layers. If you don't know what this is, google it.
When positions are being mirrored, it's because X-mirror is on. You're right that the doberman's is on unlike the others, but you can just toggle that off. it's a good feature to be aware of when posing/animating stuff anyways.
In addition, a game's Blender file isn't always super indicative of their actual game files. For example, the blender files here have controller bones (the ik bones), which are likely not in the actual engine. Or they could be, but usually assembled in the engine themselves. All that to say that the blend file =/= game prefab assets.
But yeah blenders nuts. you'll get it.
-Oh, Missing texture on moose is legit.