Hyper-V guest doesn't see the internet through external virtual switch

The Hyper-V guest has the ip address 169.254.209.59. This is an APIPA address. It's only useful for communicating with other local devices that also have an APIPA address. You need to assign a valid ip address, subnet mask, default gateway and DNS servers to the guest.

I have no idea what you're saying or asking in your comments. The virtual switch has an ip address because you've configured it to be shared with the management operating system. This ip address is solely for the purpose of accessing the Hyper-V host itself. It has no bearing on your guest connectivity. You should assign an ip address to your guest just as you would a physical machine. If your network is 192.168.1.0/24 (or whatever) then you need to configure the guest with an ip address in that range just as you would a physical machine. The Hyper-V host is not a router, it does not perform NAT, and it has no involvement in moving traffic from the guest to the actual physical network other than being a physical "conduit" for that traffic.

It was the VLAN ID. If I recall it correctly, when you created the External Virtual Switch, the VLAN ID checkbox was disabled. When I added the VM to that External Switch, the VM was enabled and set to 2. I needed to uncheck that VLAN ID on the VM network settings, and that solved the problem, the ARP tables started to populate on both ends and the host and the guest can ping each other now. They are on the same subnet of course.

Csaba Toth is correct. To make it clearer - setting "WLAN ID" is present both in the Virtual Switch Manager and under Network Settings on the Hyper-V machine. BOTH must be unchecked.