You kinda maybe got it. But let me throw out a few things that may help.
You won't have any duplicates and you won't need NoMoreDupes if you switch to IMAP (like Gmail or Yahoo Mail) or MAPI (like Hotmail, Live Mail, MSN Mail, Outlook.com). In fact, you can take your entire POP inbox, your Sent Mail items, Saved Mail folders, Drafts, etc., and (in Outlook) drag them onto your new IMAP or MAPI email account in Outlook. Bang! Your synchronized across all devices. Stop using POP mail and you're done with that outdated POP garbage forever. Your new mail account(s), inboxes, etc., will appear in the left-side Navigation Pane of Outlook.
Yes, you can set your "Reply to" email address in your Gmail account (via the web) to be your TWC address if you like-- but since you're using Outlook (which has its own "Reply to" tag, you'd also want to input your TWC address in "Sent from" in the account settings of Outlook. Also, to prevent spoof mailers, Gmail may require you to (one-time) verify that you are the owner of both email accounts.
I don't know if you care, but if you send mail in this way (with essentially a spoofed "Reply to" address), then any of your email recipients who use Outlook (and AFAIK, Outlook only) will see your "From" address as "Joe-Blow@gmail.com on behalf of
Joe-Blow@TWC.com". Outlook does this as a safety precaution to alert its users to potential spoofing.
I don't know about TWC's POP mail, but you can typically either:
1) Go into your POP mail account settings (on the web) and forward your mail to another email address (like a Gmail address). That should happen instantly. Or...
2) Go into your Gmail account settings (on the web) and have it periodically retrieve your POP mail. I think that will only happen, maybe every 15-30 minutes, so option 1 might be better if TWC supports that (hopefully without putting a "FW:" in the subject line.
POP mail doesn't synchronize so you'd really need to stop accessing your POP mail account directly from Outlook, otherwise Outlook will download your TWC POP mail from TWC's mail servers and delete it from the TWC servers so that it won't be accessible to your phone. So you should:
1) Delete your POP mail account from Outlook... or (probably better) tell Outlook to stop retrieving your TWC POP mail. (Go into Outlook's "Send/Receive" settings -> "Define Send & Receive Groups" -> "All Accounts" -> uncheck "Include the selected account in this group".)
2) Add a Gmail, Yahoo Mail, or Hotmail account to Outlook, and access your mail from the Inbox of that account.
I really think you'd be better off sending an email to your friends, telling them that you have a new email address that will synchronize between all of your devices and which you'll be able to keep even if you change ISPs. You can write the brief email, click on your address book, select all, and send. It'll take about a minute. If anybody fails to note your new email address, it's no big deal because you'll still be picking up your email from your TWC account.
You can use Outlook for calendars, tasks, notes and email, regardless of whose email service you use. I do. For example, in Outlook, I have a legacy POP email account (from my old, old ISP that I never use anymore) 2 Gmail accounts (one of which is my primary email address), a (spammable, don't-care) Yahoo account, a Hotmail account that I use for dealing with business, and a Live.com account. Each has its own inbox, etc.
In Outlook, I copied my Outlook contacts and calendar items into the contacts and calendar folder for my Hotmail account (in Outlook) so that they would automatically sync to my Android phone. (Gmail doesn't sync Outlook contacts and calendar to Android without the use of 3rd party software, so I use Hotmail in Outlook to sync contacts & calendar, even though I use Gmail as my primary email account.)
We can now synchronize Outlook & Android without a 3rd party app or Exchange Server!