I am not sure about the sent mail question as I am not at home to check my account. But, with pure POP, you can set-up GMail to delete the email once it is downloaded to your PC. POP is best for people who generally use a single computer to view his/her email or get annoyed seeing the same email every time to check their email.
So, I am not sure about your specific use case, but I am starting to believe that POP is broken at the moment.
I hate to be condescending, Brian, but you're being just as bad as the people you complain about for recommending IMAP over POP. Let's get this straight, POP is never "best" as you put it for anyone. IMAP does everything that POP does, plus more.
POP was a protocol intended for people to be able to download messages from their email server and then delete them from the server, so the only remaining copy was on the person's computer. The reason for doing this was that it meant the server needn't receive requests from the user every time they read an email, and also it means the server can have a very tiny storage for that person's emails because it only ever had to remember the contents of the undownloaded emails. All emails, after being downloaded, get deleted on the server end.
IMAP is an extension of this, meaning that all the messages always stay on the server. If you then reply to the email, the server is informed and updates a flag on the message as appropriate. This means that when you go back to the web interface, you can see that the message has been replied to. If you do use more than one computer, a change on any of them updates all of them to be in sync. If you used POP for more than one computer, they would all work, but the messages for your email would be distributed between your computers since if one downloads the messages and deletes from the server, then another cannot see that message any longer.
The reason for switching to IMAP with GMail is that the GMail servers (under normal circumstances) can handle all these requests from the users without issue and you don't need to delete emails (unless you want to) because GMail offer so much space.
Rant over ... the issue with the GMail not working in Snow Leopard is (I'm pretty sure) to do with the extra load on the GMail servers these past few days. If everyone who's installed Snow Leopard starts using GMail in their freshly installed Mail App, then they will begin using IMAP because of the automatic setup for them. This puts extra strain on the servers, and initiates a download of several gigs of emails as the Mail app syncs with the servers.
The reason I'm 95% sure it's not the Mail app itself is that the web interface for GMail is also acting up, indicating it is a server side issue. I think all the issues will rectify themselves in a while and any tinkering is completely pointless because it's not on your end. Just be patient!!