If you have a device still using a device with the old 2 step verification system but have since migrated your AppleID to the new 2 factor authentication, it should still work. However, a 2 step verification recovery key was a complex code generated once, when you enabled it. If you don’t remember that code, you’re stuck, as was indicated to you when you enabled 2 step verification (It was entirely up to you to record and remember that recovery code)
Two-step verification for Apple ID - Apple Support
And and the newer system - Two-factor authentication for Apple ID - Apple Support
There is also no more me.com - that system was disabled more than 7 years ago now. If you have a me.com address, nowadays it is nothing more than a shortcut to your www.icloud.com email address.