Can you connect your AirPods to the Apple Watch series 3?

Yes, you should be able to do so with no problem. See this for help, Set up AirPods with your Mac and other Bluetooth devices - Apple Support

Yes.

But it also helps if you have content to listen to via your Apple Watch

https://support.apple.com/en-us/HT204691

Yes. If you connect the AirPods to your iPhone it should appear on your watch as well. I do the same with my beats studio 3s