I've created a custom column in my "Inbox" that allows me to leave short notes to myself. However, when I move the email to another project folder the custom column note goes away. Looking for guidance.
You should be able to go to the "Change View" dropdown in the ribbon's Current View section and select "Apply current view to other mail folders." This will open a window that shows all email accounts with all folders/subfolders. Select the ones you'd like to apply the current inbox view to and select "Apply view to subfolders" checkbox.