I have seen many times where a folder in Thunderbird with this option set indicates that there are no new messages in it. But when you go to the Gmail web app it shows a new message in th folder. Clicking on the folder makes the new message appear.

This has been an unresolved problem with Thunderbird since version 0.3 (!).
See https://bugzilla.mozilla.org/show_bug.cgi?id=221792

The only solution to this I can see is to disregard tip #2 and do your filtering in Thunderbird. Moving a message with a Thunderbird filter will apply the appropriate tag to that message.
But then, as you say, you have to keep your filters synchronised across the different clients you use.

Which isn’t a great solution but at least no messages will fly under the radar.

Or hope against hope that it will be finally fixed properly in Thunderbird 3 (I wouldn’t hold
my breath though ;-) ).

Better use the tip from MOZILLA (http://www.mozilla.org/support/thunderbird/tips) that says:

“Check all IMAP folders for new mail
Thunderbird can download mail from all accounts when you start the program. Just open the Config Editor, search for the preference mail.check_all_imap_folders_for_new, and change its value to true.”

which only makes one (1) change instead of changing in all IMAP-folders.