My serial number for my MacBook Air is UNKNOWN

How are you trying to obtain the serial number? If you haven't already, try clicking the icon in the menu bar and then select About This Mac. Your serial number should be at the bottom of the list.

As to the question of why you're unable to sign in to FaceTime or Messages, it's hard to know without a little more information. What happens when you try to sign in to one of these apps? Are you signing in with a valid Apple ID?

See these documents.

How to set up and activate iMessages for iPhone and iPad https://www.imore.com/how-to-activate-imessage-iphone-ipad

If you can't sign in to your iMessage account or FaceTime on your Mac  https://support.apple.com/en-us/HT204408

If you get an error when trying to activate iMessage or FaceTime https://support.apple.com/en-us/HT201422. 

Use these support options if you need more help.

Get a call from Apple support to help solve your problem using this link https://getsupport.apple.com/?caller=kbase&PGF=PGF63005&category_id=SC0245&symptom_id=23362 Be patient. Sometimes phone support is not available.

Apple ID Support https://support.apple.com/apple-id

Apple Support Twitter https://twitter.com/AppleSupport  There may be a wait of an hour or so.

Apple ID account security for country-specific and region-specific help

 https://support.apple.com/en-us/HT204169

See the Apple Support article(s) below.

Serial number – Find of your Apple product

I’d contact the folks at Apple about that broken serial number: Get Support

Seems a manufacturing or repair problem...