I've had the exact same issue with my S3 on AT&T and "solved" it with K-9.
I manage a personal domain for about 20 family members. I don't have a quick or cost-effective way to move the domain to IMAP, so for now, I'm stuck with POP3. Every other mobile platform (BB, Win phone, iOS) manages this by providing a time period to sync, so you never get old messages, but for some reason, no Android client seems to offer this, at least not for Galaxy which makes me wonder if the problem is on Samsung's implementation of the APIs?
In any event, I've installed K-9. It also doesn't offer the ability to limit by time period, but it does offer a combination of several settings which seem to replicate more or less what I had before on BlackBerry.
Under Account Settings -> Fetching Mail:
- Set "Local folder size" to 1000 messages
- Uncheck "Sync server deletions"
- Turn on the setting "When I delete a message, Delete from server"
As long as you don't exceed 1000 messages between downloads from the POP server, this works as follows:
- Downloads email from your POP server to your phone, leaving it on the server
- Keeps email on your phone even after you download it to your PC and remove it from the server
- Eventually starts to remove messages from your phone as the list exceeds 1000 messages so the list of messages doesn't grow forever
As a bonus, K-9 gives you the option to store email on your SD card (Settings->Account Settings->Storage)
The stock email client is missing the critical setting not to sync server deletions, so once you download mail to your PC, it disappears from your phone. K-9 solves this problem and allows you to approximate what I think you're looking for.
Of course, if you exceed 1000 messages between downloads, you'll see the original bug of old messages coming in as unread, an issue that doesn't exist if it had a "Period to sync email" option.
Hope this helps...