You'd just need to move all email into your Inbox on webmail.
Before you begin, I'd replicate the Custom folders in webmail into Outlook. This way, when you move all email from a Custom folder to the Inbox in webmail, they'll download at one time into Outlook. Then select all and move.
To download and remove from the server, you'll need to configure your Outlook for POP.
As for addresses, you may just need to copy and paste the info in a intermediary dox and recreate in Outlook. If you're handy with Excel, you could copy-and-paste into a spread and just import into your contact folder.
www.cox.com/.../cox-email-server-settings.html
UPDATE: There is an Export option in webmail for your addresses: Contacts tab > Action for Contacts icon (3 horizontal bars) > Export > Format (CSV).