Where do these little white folders appear in Mail? Does Mail still display the
Inbox nonetheless?
In the Finder, go to
~/Library/Mail/IMAP-username@mailserver/. With that folder open, do
Edit > Select All (⌘A), then
Edit > Copy (⌘C), and paste it in your reply to this post to let me see the names of the files and folders it contains -- you may disguise any mailbox names you wish to keep private.