ok so the cloud switch may be playing some issue here.
that's the only difference between us--i did a build from scratch (this is my first samsung device and first carrier branding device in a long time) so no restoring of photos or anything and you used the switch app to retrieve or sync from a previous device to the new note 8.
well since you dont have samsung cloud setup, this may explain why the recycle folder is there, but why certain photos go to trash is weird indeed. can you "restore" these photos that are in the recycle folder BACK to their original folder?
what version app are you on? i have 5.3.07.402 and its latest build according to the app.
would you be willing to setup the samsung cloud storage? its free and this would most likely enable the "trash" folder plus "cloud view" and "image analysis".
if nothing else this would confirm the app is working as designed and the issue is likely unique to the way you setup up your phone.