Re: MacBook Air - Forgot Password & Apple ID not working?
Seems you are using MacOS 10.12 (MacOS Sierra) on your Mac ... ie: is earlier than MacOS Mojave (MacOS 10.14)
For a Mac with MacOS Mojave or earlier:
See the various options in the article below to choose what seems best for you:
How to: Change or reset the password of a macOS user account in macOS Mojave or earlier - Apple Support
Eg:
Re: "My daughter has used my MacBook Air with her own login for so long
that I can’t remember what my password is"
If your daughter has "Admin" privileges:
once logged into her account your password may be able to be reset from there (see above article)
or
If you are actually using a Mac that has a newer MacOS version installed:
For a Mac with MacOS Catalina (MacOS 10.15) or Later:
See what to do: If you forgot your Mac login password - Apple Support
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Also, if needed:
If you forgot your password, and / or Apple ID:
https://iforgot.apple.com/password/verify/appleid
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Re: "when I try to start up through the Recovery OS it asks me to enter my Apple ID & password,
however this isn’t working either due to a “network error”, but it is connected to my network!"
Network security has likely changed a lot since Mac OS Sierra, and much newer MacOS versions are mostly in use.
Eg: If you haven't upgraded your Apple ID to two-factor authentication, your account uses security questions to help keep it secure.
Otherwise: A Verification code is mostly required when using an Apple ID:
You could try logging in to your Apple ID Account online, and see if you can update your info. / settings:
https://appleid.apple.com/
What to do: If you forgot the answers to your Apple ID security questions - Apple Support
Eg: says:
“If you answer your security questions incorrectly too many times in a row, you will be temporarily unable to use your security questions. Try to reset your questions after the waiting period ends”.
How to: Change your Apple ID security questions - Apple Support
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Two factor authentication has mostly replaced the use of security questions:
About: Two-factor authentication for Apple ID - Apple Support
and
How to: Get a verification code and sign in with two-factor authentication - Apple Support
All the best :-)