How to do I access games purchased on Xbox One & using @outlook.com email on my new Xbox...

You would need to sign into the original account that purchased the games on the new console and set the console as the Home console for that account so you can share the games with your new account. https://support.xbox.com/en-US/help/hardware-network/console/my-home-xbox

However, keep in mind things like game progress and in-game items will still be tied to the original account. There is no way to transfer those between accounts.

The email address I'm using is a third party email address, yes. It is a gmail address. I'm fairly confident somebody removed my email from my original xbox account and replaced it with a new one, because I called microsoft account support and they told me that xbox gamertag was registered to an outlook.com email, and I have never used outlook.com. So either my xbox account was hacked and stolen or something messed up within the system and the email was changed to a microsoft email that I do not have access to. The only thing I have proving that it is my account is the various emails sent to my original email on the xbox account verifying that it belongs to me. I had a lot of stuff on this minecraft account and I really cannot afford to throw that all away and buy a new one. I checked a previous email that I used with the account, and there was nothing on it, since I already transferred it to my new email before it got unlinked from that too. I could not find any emails about an email change on the xbox account.

I have a feeling that I know the answer to this but I am taking a shot in the dark here.

I have an Xbox Live Account that is associated to a Gmail email address. I recently purchased a family subscription to O365 for a year and got a new Outlook.com email address. My original Gmail address account has a ton of purchased games and saved game
data on it that I want to move to the new Outlook.com account. I know you cannot move the games or data, but I would think that I would be able to associate the existing account with the new Outlook.com account, but it seems the new account already has an
Xbox Live account and auto-generated Gamer Tag associated with it so when I try to add the Outlook.com alias to the Gmail Xbox account I get an already exists error.

Are there any options here to accomplish my goal? I want to get rid of the Gmail account altogether and just use the Outlook account for everything without having to purchase my games again and lose my saved game data. Can the Xbox profile associated with
my Outlook.com account be deleted so I can add the alias.

I get the Microsoft doesn't want people selling gamer tags and saved game data, but they should also assist people when they are trying to use their products and purchase their subscriptions.

Any help is appreciated. Thanks.

You should be able to change the email address associated with your gamertag. You can do that here:

https://account.microsoft.com/profile/

Click the "Manage how you sign into microsoft" and in there you should be able to change the primary email address associated with your accounts. Set your new email address as the primary. Then to double check you can remove the profile from your console
(not the games or save data) and just re-download the profile and make sure you use the new email address to make sure it correctly recognizes it.

If I transfer my XBOX Live account to my new email, will the games I purchased be associated to the new email as well? How will I play the games I purchased using the old email?

Thanks in advanced.

Only one MS account ever. That change was when outlook.com became available. Everything is listed under my outlook.com email address. As mentioned, no other games have had issues. I have had multiple Xbox 360s, multiple Xbox Ones and have always used the same account and have never had any issues downloading games whether new or old except these two recent additions to the Back Compat library.