Setting up 2-factor authentication, I never had the opportunity to enter my verification code, yet now, 2-factor auth. is turned on (I went to get it, and when I got back to settings, it was already on)! Don't I need it to add another device?

See the part below about Apple's discretion. I think Apple exercised its discretion and turned it on. But, you should have been notified.

See Two-factor authentication for Apple ID https://support.apple.com/en-us/HT204915  See this section especially Setup two-factor authentication for your Apple ID. From this section, "Two-factor authentication is available for Apple ID accounts with at least one device that's using the latest iOS, iPadOS, or macOS. Learn more. You can follow these steps on your iPhone, iPad, or iPod touch to turn on two-factor authentication." 

And, Use the “learn more” link above to see more about the Availability of two-factor authentication for Apple ID https://support.apple.com/en-us/HT205075. From this link: "While most iCloud users can now turn on two-factor authentication, certain account types may still be ineligible at Apple’s discretion."

Information only.

If you recently updated your account, however, you can unenroll for a period of two weeks. Just open your enrollment confirmation email and click the link to return to your previous security settings.

Apple ID -Two-factor authentication.