How long ago did you get the new dog? Do they get along? My Nicki acted a lot like your Hayden after a few serious fights with my husband's dog. Perhaps they got into fights without you being aware. Nicki also learned to raid the trashcan from Sammie -- something she never did prior to sharing her home with a second dog.

It could be that Hayden poops ion the closet as a serious way to mark his territory so the new dog knows that is HIS place. He could feel safe in the closet and doesn't want to share it. Crating is a good thing because most dogs take refuge in their crates when stressed or anxious, it seems as if he has chosen the closet as his refuge.