Why am I unable to access my Apple Account consistently?

Enable two-factor authentication. That blocks access without the password and access to a trusted device, and (after two weeks’ enabled) that also completely removes the security questions. You’ll want to confirm and set a trusted telephone number or two, trusted email addresses, and potentially also a Recovery Contact, when you switch over.

I’ve chased a few of these cases on various services over the years, whether due to password corruptions in iCloud Keychain or otherwise, or due to apps such as mail accounts running with old passwords. Though those usually show failures, which you’re not reporting here.

I’ve also met lockout cases due to caps lock, keyboard language settings, add-on anti-malware issues and failures, and other mundane causes.

If you’re using third-party add-on anti-malware on Microsoft Windows, try switching to the built-in Windows Defender anti-virus, or whatever Microsoft is now calling that. Anti-malware is a good potential culprit when things go weird in any platform. The stuff is arguably malware, with a EULA.

The free Apple Music offer uses whichever Apple Account you’re using, yes.

If you can't reset your security questions and don't have two-factor authentication enabled, you're headed for account recovery again, or have lost control of the Apple Account.

Your Windows install or something else in the local hardware or software is likely corrupted, or is flaky, given all you've been through.

I'd start by entirely removing all add-on anti-malware add-on apps per the vendors' instructions, but I'd fully expect to need to reinstall Windows and apps here, given the persistent issues reported, and given many previous suggestions have clearly failed.

Back when I was running Windows with whichever corporate-mandated add-on anti-malware was then in corporate favor, I'd usually have to reinstall once or twice a year due to something somewhere clobbering something else. I hope that's all gotten better since, but the corruptions were an immense source of frustration for me.