Mac Mail deletes messages after 24 hours

Mac Mail started deleting my messages after 24 hours 6 days ago. I believe this is a OSX Lion issue as I run 10.8.2 on my main machine (iMac 27 Dec 2010) but I also have an old iMac G5 (2005) running OS X 10.5.8, and on ths machine my messages are not deleted. An additional problem is that new messages to all accounts remain marked as unread for less than an hour on the new machine but remained marked unread until they are read on the old machine. My sent messages are not deleted.

I use a private email address from my own domain for all of my emails, except for a very few on BTinternet and Yahoo adresses. My webmail for my domain also is having emails deleted after 24 hours. I have spoken to the domain host customer care (very helpful) but they cannot spot the issue. And as it doesn't happen with OS X 10.5.8 it seems to me that it cannot be either the host or my current provider BT. However, I have only switched to BT recently and this deletion problem only started once their service was fully up and operating ( which was a long and messy process leaving me utterly disenchanted with BT).

I also have an iPhone 4S using iOS6 which doesn't delete but also doesn't show new messages as unread on my private domain address. On the iPhone new messages to my BT and Yahoo accounts show as unread.

I have time machine running with an external hard drive so I can see old messages there but thats a laborious procedure and the whole thing is doing my head in. Anyone got any explanations or, better still, solutions.