If the Apple Watch has wifi - does it connect to my local router? It would appear the Apple Watch has the ability to communicate via wifi. 

When I set mine up, I didn't have a step to identify my local router, so I assume it may have obtained configuration about my local router from my phone, or that it doesn't connect (directly) to my router at all. 

My question is: If the Apple Watch has wifi - does it connect to my local router?
Yes. It uses the wifi networks already configured on your phone. Mine worked over my home network perfectly with zero configuration on my part.