What is the difference between a Mailbox and a Smart Mailbox?

Smart mailboxes are basically a convenience for creating a view of some mail based on some criteria.

For example, suppose you have a bunch of mail rules that moves mail out of your inbox into other mailboxes based on the senders address. Going into each of those mailboxes to read the mail would be tedious, so you could create a smart mailbox for "unread mail" that creates a view of all the mail not marked as read.

Actual storage is based on mailboxes. Rules operate on mailboxes. Smart mailboxes only provide views. You want mailboxes. Luckily, deleting a smart mailbox doesn't delete any actual mail, since it's just a view.

Dunno. The "move" command in rules is supposed to move mail, not copy it, and does so in my rules. Since I don't know anything about your mail setup or rules, I have no way to tell what the problem might be.

Could be the mail account isn't set up correctly (is it iCloud, or something else). Could be that your mail server (if not iCloud) doesn't conform to standard IMAP protocols. Could be that your rules aren't written correctly. Could be any of a number of things.