Why does Mail keep showing that I have incoming messages, when I don't?

Question

...usually hundreds via the "burning cigarette"!

Answer 1

Quit Mail. Force quit if necessary.
Back up all data. That means you know you can restore the Mail database, no matter what happens.
Triple-click anywhere in the line below on this page to select it:
~/Library/Mail/V2/MailData
Copy the selected text to the Clipboard by pressing the key combination command-C. In the Finder, select
Go ▹ Go to Folder
from the menu bar. Paste into the box that opens (command-V), then press return.
A folder window will open. Inside it there should be files with names as follows:
Envelope Index
ExternalUpdates.storedata
Move those files to the Desktop, leaving the window open. Other files in the folder may have longer names that begin as above. Move those files, if any, to the Trash.
Log out and log back in. Relaunch Mail. It should prompt you to re-import your messages. You may get a warning that the index is damaged and that Mail has to quit. Click OK. Warning: The process may take hours if you have gigantic mailboxes. For reasonable-sized mailboxes, it should only take a few minutes.
Test. If Mail now works as expected, you can delete the files you moved to the Desktop. Otherwise, post your results.