Will turning on two-factor auth require me to know my security question answers? (because I dont)
Turning on two-factor-authentication may require your security questions.

You might be asked to answer your Apple ID security questions.

https://support.apple.com/HT204915
Once 2FA is activated, your security questions are deleted.

Do I still need to remember any security questions?
No. With two-factor authentication, you don't need to remember any security questions. We verify your identity exclusively using your password and verification codes sent to your trusted devices and phone numbers. When you enroll in two-factor authentication, we keep your old security questions on file for two weeks in case you need to return your account to its previous security settings. After that, they're deleted.

https://support.apple.com/HT204915
Therefore, I suggest you attempt to set it up, since the use of ‘may’ in the first quote means there is potential that the security questions are not asked for, and it you are successful in setting up 2FA your security questions are removed. However, this is on the condition you have a trusted device to receive the authentication prompt, which requires you to be logged in to your Apple ID on the device.