The fact that there are no errors during install is strange, since the Core installer is supposed to be removing the App_Code files for you during that process.
My testing was done in 4.7 (the most recent Core version at the time), and I did check all the upgrade paths there, but it's possible a change in 4.8 caused a bug. I'm setting up a 4.8 test platform anyway, so I'll run some tests to see if I can reproduce the issue as you described.
You should be able to fix this issue right away by restoring the 5.0 DLLs you deleted (assuming you made a quick backup first, otherwise you can just take the DLLs from one of the download packages) and following the instructions in DesktopModules/Reports/AppCodeCleanup.txt to remove the App_Code folder. I would recommend moving the folder, rather than deleting it, just in case that doesn't fix the issue. If it works, then you should be able to delete it.
Hope that fixes things, if not, let me know and I'll do my best to help!