No a soul:

Why Yu and then Wukong could lift it and why others couldn't?

I have basic knowledge of Yu. Could you tell us more about him please?

I know that it is said that he was guided to his future wife by a fox spirit or that his wife was a fox spirit.

This is not like Mjölnir (from Marvel Comics' Thor), which no one can pick up unless proven worthy. The staff is just heavy. Wukong can pick it up because he is strong. Yu the Great was likely strong, too. There's nothing else to it.

I'm not an expert on Yu, so I can't explain much. I recommend reading his Wikipedia article. But beyond that, some sources say that he and his son, Qi, were born from stone. These same sources also call his wife the "Girl of Tushan" (i.e. the "Girl of Earth Mountain"). She is thought to have been the primordial goddess Nuwa. I discuss this in the following article.

Jack still has that Monkey Staff-it’s part of the items Hunter blackmailed Jacks professors into giving back

Monkey-brain there REALLY likes how Hunter smells, and Macaque thinks full-Monkey-Jack is hilarious, so he doesn’t interfere when Jack gets like that

Bonus under the cut: