After much wrestling with pivots and centers, I've finally managed to get my models in the Blender Game Engine behaving as intended... well sort of.
I'll leave my comments/rants about pivots and centers for another day. Suffice to say, it was an unplanned learning curve.
This is the final product.
CuteTobo Blender Collision Test 20150220-001 - Indie DB
Ok so not very exciting, however it's something.
Progress is being made. Hooray.