You have 2 different problems. The DDNS is pretty straight forward if it is not setting the value to the WAN IP it must be a bug in the router firmware. There is almost nothing you can set in these configurations other than where the DDNS server is. Make very sure you actually have a public IP you should be able to see the IP in the wan setting on most routers. Otherwise I would try a firmware update.
The second problem is pretty much there is no such thing as Wake on "WAN". You are attempting to use a hack with the port forwarding and it has massive restrictions. The main one is that the ARP entry in the router that maps the internal IP to the mac address of the machine you want to wake will time out. When this times out the router will discard all data. I have yet to see a router not running third party firmware that you can set static ARP entries.
Your best option is to just use the low power options modern pc have. The difference in power used from a machine being in low power mode and one in WoL mode is very small.
If you insist on using WoL your best option is to get a router that you can remotely access and have it send the WoL packet for you. Since the router is actually on the lan it can send actual WoL packets rather than the hack packets you get when you port forward. Asus routers have this remote WoL feature.
You can also use things like raspberry pi that you leave active on your network and request it send the WoL packets. But again much power does this really save.
There are also power controllers that you can get that allow you to turn power on and off. You just set the PC to boot as soon as it gets power.
I thought the problem would have more to do with the router setup itself. My modem and router are on two different subnets. The modem is the one that recieves the WAN IP. It also assigns the router a network address of 192.168.0.10, which is the IP that is sent to the hostname.
I did enable IP passthrough on the modem since I have all the radios turned off, but I guess the router doesn't actually recieve a WAN IP? Would enabling bridge mode or transparent bridge mode on the modem fix this issue?
The Archer C7 does have an ARP Binding mode that I enabled along with a static IP to the target computer.