Actually SENDING is done via SMTP regardless of the way you receive email. I didn't think the servers could tell if you are mobile or not for email purposes. Even on a web server they can only tell based on the headers that the browser sends, which can easily be spoofed with the right browser. If that is documented by Charter, then so be it, but it sounds very strange that they would differentiate that way between IMAP and POP3. It sounds more like the problem is with how the SMTP server is getting its authentication. Some SMTP servers rely on logging in with a password each time. Others require you to authenticate by checking mail before trying to send (typically within one to five minutes). So you might have to tinker with the outgoing authentication (try clicking the outgoing authentication off and just make sure you do mail check immediately before sending and see if that works. If so, then it means that the SMTP relies on the POP3/IMAP authentication to validate the user.
You called it right about deleting. If you are using IMAP, then deleting on the tablet will delete on the server next time it syncs so you would not want to delete it on the tablet. It will delete itself after you download the mail on the PC, assuming the PC deletes the mail on download. You can tell pop3 not to do that since pop3 is not actually "synced". Pop3 essentially pretends to sync by downloading the email and respecting your server choices (never, when deleted from inbox - strangely enough delete on download doesn't seem to be available on the tablet client).
If the issue is that you want to keep certain emails on the tablet after they are deleted from the server, you have a couple work arounds. IMAP allows you to create additional folders so you could create a "Tablet" folder and move the mail there that you want preserved on the tablet. The beauty of this is, that folder would still be on the server so you could still get to them from the PC as well if you set up an IMAP account on the PC. Or you could forward it to a secondary account that is just for the tablet. Email syncing on multiple devices is a little tricky to get exact but its more about workflow than about the email client
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I personally do use IMAP on the tablet and POP3 on the PC so that the PC cleans out the email box regularly. But as I write this and think things through, I"m starting to think that with today's trend toward cloud computing, maybe just using IMAP everywhere is a better approach since IMAP is essentially a cloud based solution. Though THAT depends on how much space you are allowed on your email server.