Well I figured out part of it. When I copy a system defined view, like "Sent To," the view copies to the IMAP folder, but it does not "stick" to the IMAP folder until I go to that folder, click on "Manage Views...," and save there with the same view name.

Now I have a "Sent To" view for both the POP3 folder and the IMAP folder. However, the user defined views simply will not copy from the PST folder to the IMAP folder.