First question is are you really talking about Good Neighbor Hotels? There are lots of them and they get NO perks from Disney. While I don't know for sure I don't know why they would show up on the app.
Yes, if you book via MD, then yes, that will appear in your MDX however, if you book directly via the hotel, you may have the option to add your confirmation to MDX and see your reservation in MDX but I'm not sure. I know you can do that with Swan, Dolphin and Swan Reserve but have not tried this with other properties.