I haven't used pop mail in a while, but I recall that I'll want to set either the phone or the computer (or both?) to leave copy of messages on the server.
If she wants her e-mail on both, then
both devices need to leave messages on the server. The moment an e-mail is downloaded off the server, it's no longer available to any other device.
You'll also need to check her desktop. If it's been downloading e-mail off the server all this time, then she's not going to see any of those e-mails on her iPhone. The messages reside only on her desktop, and that's that.
This is why for multiple devices, POP is just a bad idea.
Bear in mind that this probably means her inbox is going to reach quota pretty fast. ISPs just don't like to give people much space on their servers for mail, leaving the job to gmail, yahoo and private label companies. Heck, some ISPs don't even bother, and contract with Yahoo or Microsoft Live directly or just tell you outright to sign up with them on your own.
She may not like this, but realistically, if she relies on her e-mail a lot, maybe it's time to switch to one of the web-based services that can offer IMAP or Exchange-based sync. They'll do a better job, many of them don't cost anything, and her messages will probably be more secure, anyway.
And if she doesn't want her address to end in yahoo.com or gmail.com, yahoo does work with AT&T to give their ISP users e-mail... and you can take advantage of this too. If you go to
http://att.my.yahoo.com, you can sign up for an e-mail account that has an att.net domain, and you don't even need to be an at&t customer.
What about sent mail? I assume that if she sends a message or replies to an email from the phone, that a copy will not end up in her sent messages folder on the PC. Is that correct, or are there ways to set this up so that the two are essentially synced to each other?
It's possible to set up the Phone to upload sent mail to the server where it will keep a copy. Again, all of the above caveats apply.