Hi Old-dear!
Officially, yes. A 2013 MacBook Pro cannot officially update past macOS 11 Big Sur.
Excel/Microsoft 365: Microsoft only supports the three most recent versions of macOS. This OS is too old, and Excel cuts off access and updates.
Apple Mail: Mail likely stopped updates because email provides (like Gmail, Yahoo, or Outlook) updated their security protocols (like OAuth 2.0) which the older version of Mail in Big Sur can no longer handle.
As for deleting that file, there could be a number of things causing it. The safest way to delete the file would be to Delete it in Safe Mode.
Turn off your Mac. Turn it back on and immediately press and hold the Shift key until the login window appears. Try deleting the file now. Safe Mode clears system caches and bypasses certain restrictions that might be locking the file.
Also, check your storage. If your Mac is completely full, it will often refuse to delete files. If this is the case, move a few files (like downloads) to a USB thumb drive to free up a bit of space, and then attempt to delete the file.
Hope this helps!
-Owen