I'm using Thunderbird to download mails from a Hotmail account. It works all right, except that the mails in the server are only marked as read when I log in to the webmail service and read them there. Thunderbird internally does mark them as read, it's just that the server ignores that. This works fine with my GMail account.
With POP3, there is no way to tell the server about local state (read/deleted/etc). It can only go in a single direction - from server to your computer. So the server can't even know.
Basically, everything POP3 can do is: get number of messages; get a list of message sizes; download a message; delete a message from server.
(Some POP3 servers do mark messages as read server-side once you download them over POP3, but this is not very reliable, and many dislike it.)
IMAP4 (the one you use with Gmail) can do much more - folders, downloading headers only, search, sometimes even shared inboxes.
In gmail go to