Walking deforms the neck and pulls down the body

Hi, I am having some trouble understanding the problem, but I asume what you mean is that the head deforms the body whenever you rotate the head. If that is the case, you can simply go to your properties window (make sure you're in rig mode) and find the option named "independent." There will be a checkbox below it. The option "hinge" worked for me when I had that problem. Below is a screenshot of the option.

I noticed the "slouch" too on puppets at times. Best I have found is to make sure the "head" and "neck" handles are far enough apart. You have them on the same handle at present, which is not a good idea. The relative position of handles can change warping. I generally go for "Neck" as a handle at the base of the neck on the artwork.

Try turning on the yellow-mesh-of-triangles icon at the bottom of the scene window. Sometimes you can see the mesh distorting badly, giving a hint what to try.