It shouldn't be doing that (at least, it doesn't happen with any of the accts I have configured).
Are you using IMAP? POP? Exchange?
If you're using IMAP and you delete a message from a different client (say, Outlook or webmail, etc.) It will no longer appear on your Droid. (IMAP only looks at what is on the server; if you delete it from one client, it will be gone for all others.)
POP can either pull messages off the server down to your client, making them appear "gone" for any other clients you use, or leave them on the server, making them available for any other client you have configured. If another client is set to pull messages off the server, they will be "gone" from your Droid. (If you do have another POP client set to pull your messages off the server, and you always check this client around the same time every day, it may appear that messages on your Droid are mysteriously deleted around the same time every day.)
With an Exchange connection, you can also delete messages with another client, making them appear to "vanish" from your Droid.
Do you use other clients to check the same account? Or just your Droid?
Hmmm, not a known issue to may. As asked already, you need to be a little more specific on what email you are using. That being said, I recommend going to K-9 for your non exchange email purposes. Still has some bugs to be worked out, but so much better than stock email apps IMHO.
Just because you don't see older emails in your inbox doesn't mean that they were deleted. Depending on your settings, you may only have your Email program setup to display emails from the last 1 or 3 days. So you may want to double check that first.