Will separate dishwasher drain cause sewer gases and wet dog smell?
The smell is almost certainly from your AAV.   While they are not to dispose of gases from the pipes, they always do.   Having an AAV makes really no sense because you are going into a wall.
It doubly makes no sense because there seems to be another drain without one - unless I am not seeing it in the picture.
The only &quot;fix&quot; is to have this drain attached to a proper vent which is probably in the wall behind it.   There is no other &quot;fix&quot; unless you just upgrade the AAV and hope it does a better job of not letting smells out.
As far as dishwasher - it should be going into garbage disposal.   I don't think that is the smell issue but might as well get it right.