WH, Your attorney is doing the right thing. If you let this slide, he'll continue to do such things. Just as a child acts out, you need to correct them right then or they won't understand later on why they are being punished.
Now, about the dog and the note. In the future, your h is responsible for those things. He doesn't have the "right" any longer to tell you what to do. If a note needs to be written and it's on his watch, he needs to do it. He needs to find a way to ensure that the dog is let out, whether it's to give a key to the neighbor or put papers down...but that's his responsibility right now. You two are no longer together and the sooner he realizes that, the better.
I know, you think you have to be at his beck and call w/the children, but he's an adult and the amount of time it took for him to call you about the note, he could have written it. Stop being his mother and taking care of everything for him...he wouldn't do it for you. You are separated and heading for divorce...drop the rope and allow him to experience what it's going to be like with full responsibility for all of the child, pet and home issues. You are no longer responsible for what goes on over in that house.
Drop the rope! Your attorney is doing the right thing. So what if your h gets mad. He brought it on himself for lying, cheating and yes, stealing from you and the kids.
Sit quietly, the answers will reveal themselves when you least expect them to. The past is gone, the present is a gift and you need to focus on today, allow the future to reveal itself when it is ready.