xbox Live account is attached to my dads email and his xbox live account is attached to mine

I would suggest contacting the support team, but I don't think they can actually help with this

if they are Microsoft email addresses then they are permanently linked to the Xbox account.

if you add an alias email then remove the Microsoft email address it will delete the Microsoft email account and you will no longer be able to access anything else connected to it, such as Office 365, and you will no longer be able to send or receive emails
on that account

Hi there,

You can change the email address associated with microsoft accounts by following the steps here. From these
steps you can designate a new prime alias for your account, and it will replace the email address that is being deleted (talk talk).

If only that was the case. I did not lose my email that was attached to the Windows Live ID/Account, I've lost/deleted the Windows Live ID/Account as a whole. This is resulting in error status code: 8015D080 because the Gamertag is no longer linked to an
active Windows Live ID/Account. My question is as to whether I can attach a new email/Windows Live ID to the Gamertag.

No. One profile isn't attached to an email. Its an offline account. The other is my new live account