Andy, thanks for this. You posted this ages ago, and apologies for the long delay. My limited computer brain has only just started to take in what you were saying.
Yes, my Thunderbird server setting on my current gmail email address is imap.gmail.com. On the old freeserve address is was pop.orange.net. It worked fine on that, but then it also worked fine on the imap one for quite a few months. As I understand it the setting is that deleted emails are meant to go to a deleted folder, but recently they have gone to 'bin'.
I'm not sure how I would change to pop3. Would I just type something in the setting for 'server'? But I'm a bit scared that I would mess the whole thing up! Any further advice from you if you read this, or anyone else, would be appreciated.
If the account has been setup on thunderbird as IMAP (i suspect it is) then changes in thunderbird will take effect on the mail server (deleting and sent items take effect on server too). Each mail provider have different policies/rules/etc for handling 'deleted' emails. What you want to do is set it up as a POP3 account.
If its POP3 already, go to account settings in thunderbird (under tools i believe) and look for settings relating to trash/deleted you want to uncheck the likes of "delete trash on exit" and "delete from server after x days/months" or something to that effect.
Setting it as pop3 will not show any sent emails or ones marked as important if you login to webmail to check emails too. Its all downloaded locally so no changes made on server.
Alternativley to keep it as IMAP and benefit from its features then set up a new folder/category on thunderbird call it archived or bin or something unique and just move unwanted emails to there till you decide to delete yourself.