To recap:
For POP accounts, under each you@example.com access point in the mail panel, you should just have an "Inbox" view and a "Sent" view. There shouldn't be a "Trash" view.
The "Sent" view for each POP account access in the mail panel will show sent messages for its account. But, you need to make sure "Show sent" is set for each sent view.
There's also an "All Messages/Sent" view which shows sent messages from all accounts. But, "Show sent" has to be set for it too to see sent messages in it.
*If* you're not seeing a sent message in a sent view, try turning off threading for that view to see if it shows up. There's a known UI glitch with sent messages that don't have any replies being hidden in threaded mode.
Now, when you del a message, it'll be marked as deleted. Deleted messages for all accounts are shown in "All Messages/Trash". But, you have to have "Show Trash" set for "All Messages/Trash". Also, for all other views like "Inbox" and "All Messages/Received" etc., if you don't want to see deleted messages in them, make sure "Show Trash" is unchecked for those views.
The show settings for each view are found by clicking on the "Settings for this view" icon on the message list toolbar for the list in question.
So, it sounds like you have "Show Trash" checked for "Inbox" for one. That's probably why you're seeing deleted messages in "Inbox".
Now, assuming things are set up right, you reset the mail panel (by clicking the wrench icon on the mail panel toolbar and choosing "reset mail panel to defaults"), and you still don't see "All Messages/Trash", maybe the "Trash" folder itself is hidden. For this, right-click on the "All Messages" access point header, customize and make sure "Trash" is checked. Then, make sure "Show Trash" is set for it.
If you still don't see the "All Messages/Trash" view, there's probably a problem with index.ini in the mail folder. You can close down Opera and edit it.
Make sure you see something like the following:
[Index 6] Name=Trash Id=6 Parent Id=1600000000 (Points to category with that id) Mirror Id=0 Type=2 Flags=259 Special Use Type=7 Keyword= Account Id=0 Model Type=0 Model Age=5 Model Flags=511 Sort Type=7 Model Group Type=2 Selected Message=0 Update Frequency=10800 Last Update Time=0 File=m2_trash.idx Skin Image=Mail Trash Search Text= Search Field=0 Search Current Message=0 Search Start Date=0 Search End Date=0 Search is Regexp=0 [Category 0] (The "All Messages" access point) id=1600000000 open=1 visible=1 unread=2If that doesn't help, there's probably something messed up with the indexer files in the "indexer" folder or the omailbase.dat. The database gives an id for each message and the indexer files tell Opera what messages (by id) should show in each view. If things are messed up here, you have a corrupted mail folder.
You can try https://www.reddit.com/r/operabrowser/wiki/m2tips/trigger_database_check to fix simple database errors. It *might* help.