Q: How can I change my rescue email address? I can't recover my security answers because apparently my rescue email address no longer exists.

You won't be able to change your rescue email address until you can answer your questions, if you don't have access to the email account that it's currently set to then you will have to contact Support in your country to get the questions reset.

Contacting Apple about account security : Contact Apple for help with Apple ID account security

If your country isn't on that page then try this form and explain and see what they reply with : https://ssl.apple.com/emea/support/itunes/contact.html

When they've been reset you can then update your rescue email address for potential future use : Manage your Apple ID primary, rescue, alternate, and notification email addresses

Or if it's available in your country you could change to 2-step verification : Frequently asked questions about two-step verification for Apple ID