One of the issues that I mentioned earlier was that the iPhone didn't access the subfolders of my inbox on my IMAP mail server. It turns out that there is a way to fix this.

In the mail settings, after clicking (tapping?) on the account in question, there is an advanced option where I needed to put in "INBOX" as my IMAP prefix.

Works perfectly. Thanks to superg on MacOSXHints.com for this tip.