Do you have Outlook set to delete from the server? Is there anything in the IMAP mailbox on the computers?
After setting up the IMAP account in Outlook, you should remove the POP3 version of the account from outlook.
1. You don't need (and shouldn't have) one account configured in Outlook as both pop3 and imap. It just creates confusion.
2. I think what you were seeing wasn't what you thought you saw. Deleted items will be removed from the server (if configured to do that) so what you saw wasn't anything unusual.
3. I don't think you need to enable anything to get imap (I didn't) - its default for mobile access. You just need to enter the imap server names in outlook.
4. This behavior looks like Outlook is deleting mail from the server after downloading it using POP3. That would account for it disappearing from the iPad too - it's deleted from the server and imap is a server-based account type, where the device has an identical copy of the mailbox.
5. you'll need to use the iCloud to sync calendar and contacts. Syncing calendar and contacts doesn't use imap - although when you set up the iCloud it will probably add an imap account to outlook for your apple id. (It did for me, but my apple id is a me.com account.)