I have a qmail server that will not let me create folders under anything except Inbox. My dovecot+postfix lets me do it.

When I create a folder it says:
The IMAP server refused to create folder. Invalid mailbox name.

Qmail is a bit of a mystery to me. It was installed by plesk, but I can't find configs for it anywhere. Mail works fine and all, but it is a bit annoying to not have Sent, Trash, Drafts, etc.. folders.

At a first place, why do you want to use qmail, as you have already postfix as MTA?
Regarding qmail, there is the command maildirmake that is used to create a Maildir in a user's homedir with the structure used by an imap server.
First make sure that .qmail (if exists in the user's homedir) or /var/qmail/control/defaultdelivery (the location may vary depending on the way qmail was insalled) contains "./Maildir/" (without the quotes).
Then you can run:
More details on qmail you can find here.

Regards

I wish I could, but we are using Plesk, and it seems to support only qmail. Go figure.

It creates the Maildir folder in /home/mailboxes/<domain>/<username>/Maildir

The problems is this. On dovecot+postfix the user's client see's this:
Inbox
Sent
Drafts
Trash

This server seems to put it this way:
Inbox
-Sent
-Drafts
-Trash

I hope that makes sense. In other words, rather than the other folders being created at the same level as the inbox, it makes them subdirs of the inbox. I don't know why it is doing this. Is this specific to qmail or do I just have it configured incorrectly? (maybe a Plesk related issue?) It seems to assume that Maildir/ is the inbox unlike my postfix+dovecot that has a .Inbox inside the Maildir/ direectory.

On the server side(postfix+dovecot) Maildir shows:
.Inbox
.Drafts
.Trash
cur
tmp
new

On the server side (qmail) Maildir show:
.Sent
.Drafts
.Trash
cur
tmp
new

In Maildir format only the 3 subdirs (cur, new, tmp) are created when you first add a user (that is what maildirmake does). The rest is created by the client used to read/write emails.
There is no folder named .Inbox. The real INBOX (incoming mail) consists of the 3 above mentioned subdirs.
So regarding your 1st post:
I don't see a problem

Yes, I understand how Maildir works to a point. I know the client is supposed to make the folders. The problem is, that it cannot! I can manually make them in the Inbox folder. This is not default behavior of any other IMAP server I have used. I'm trying to determine if this is qmail specific or do I have something configured incorrectly.

What client are you using? Are you sure it's not a permissions problem, specially if you use a web based client?
According to your previous post there is no Inbox folder in qmail's Maildir structure
So, where exactly are you creating these folders?

I have tried Thunderbird, OE, and windows mail (vista). None of them make the folders. I can make them manually, but the clients fail to make them automatically. The owner is popuseropuser. This looks specific to Plesk setup.

Sorry but I don't use Plesk, so I cannot help further. Maybe you have to contact Plesk support.

It turns out this is not Qmail specific, it is courier-imap specific.
http://www.courier-mta.org/FAQ.html#namespace

To translate that to normal speak, "The client should support us, not us support the client!"

I did find the fix to make it look "normal".
Make sure to look for the setting in your client called: Root folder path

If you put INBOX in there, it will look normal.