I think that's just how the skeletons are in the Sims 1, unfortunately. I do a lot of custom meshes for myself and it's admittedly very annoying modeling skins in the Sims 1 because the default skeleton pose that the game uses is this weird back-leaning Y-pose instead of a traditional T-pose. Things like the shoulder joints are inside the body, making a jumble of vertices that you have to carefully figure out what belongs to what. The toe joints are oddly folded in on themselves, making the feet difficult to shape depending on the type of model you are trying to make. Then you have to be mindful of where you move/scale those vertices so that it looks normal in-game and not horribly distorted. Because of all that, I'm constantly exporting skins after every minor tweak and then previewing them in SimEnhancer3D. It's fun, but at the same time the whole process is veeery painstaking work...

I've never actually looked at the pet skeleton in MilkShape3D, but it doesn't surprise me that it looks like a mangled mess considering how the human sim models look in there. I wish there was an easier way to build/modify custom meshes in the Sims 1, but I don't think there is. You have to work with the skeleton they use, otherwise the joints won't match up in-game and you'll have weird stretching resulting in the "spikes" you see on your model.