Just do (almost) nothing. All your cable comp email is already on your computer. Keep it there as an archive. When you change providers unselect that cable comp account so your email proggie doesn't search or send via their server.
You'll get a new email addy from the new cable comp. You will need to set it up in your email program (Outlook/express/thunderbird/etc)

No. When you get your new addy from your new provider you need to use that one as default and use their pop3 and smtp server.
You do that in outlook express>>tools>accounts and then you unselect your old provider's account from being used to check for mail and of course select the new provider's account as default send and receive. Don't remember exactly anymore where but should be on the same page where you setup or edit your accounts.
And yes, your emails are on your computer. Search for *.dbx files to see where they are stored on your machine.
Hotmail I always used MSN's web interface and not outlook. I don't think you can incorporate that into OE.
If I were in your situation I'd open a Gmail account, set it up as your default email in OE and start forwarding from old cable comp account to Gmail account, and send everyone in your mail list a notice that your account will change to Gmail when you quit your cable comp. Then use your new cable comp's email as just another address but keep the Gmail as default forever and ever.