Apple IDs never stop existing. While the E-mail address associated with them may stop working or be cancelled. the Apple ID remains and can still be accessed with the proper password.
It is always highly recommended to update the email address for an Apple ID when it stops being functional.
If you do not remember the Password for the Apple ID and you had no recovery email in place for it outside of the one that no longer works, then you will need to contact Apple Support and attempt to recover it with them.
Fill in the form here to Contact iTunes support for Account Security.