This really just sounds like a problem with the PC mail client software. My guess is that it is currently set to only download "new messages" or "unread messages", but once the user has accessed a message from the droid, it is now appearing as a "read" message within the gmail servers, and therefore the PC email client is ignoring it.
I don't use a client so I don't know if this is a POP3 issue or IMAP issue, or both.
To test this theory, Turn off the phone (and leave it off till your done with the test) and send yourself a message from some other email account or have someone else do it. Then try to access your mailbox from your PC. If I'm correct your email client should be able to retrieve the messages, because nothing would have marked them as "read" yet.
Also, you still have not told us important details, such as which software you are using to access your gmail account from thephone AND from the PC. IMHO You should only be using the actual gmail app from the phone and the website when at the computer. I've been doing this for years across multiple PCs and phones and have never had the problems your are describing. I suppose the only reason you would ever want to download your messages to a PC email client is if you are requiring the ability to read and respond to mail while offline. If this is not a requirement, then why bother with an email client?