Cornelicorn

It should not be very difficult: Install docker and check if its working using https://docs.docker.com/docker-for-windows/#shared-drives then just edit the commands 
docker run \ --restart unless-stopped -d \ --name steam-cache \ -v {Path of Steamcache-cache-folder on the host drive}:/data/cache \ -v {Path of Steamcache-log-folder on the host drive}:/data/logs \ -p {Main IP for Steam Cache}:80:80 \ steamcache/generic:latest  
Adjust the other commands accordingly, you can configure multiple IPs by going into the Control Panel -- Network and Internet -- Network and Sharing Center -- Change adapter settings -- right click on your Network Interface and click on Properties -- select Internet Protocol Version 4 and choose use the following IP address -- enter your main IP of your system -- enter your preferred DNS-Servers -- click on andvanced to configure multiple IPs: just click add to add another one, if you don't know what to enter as Subnet mask just enter 255.255.255.0. As you said this is one of your Home Servers let the Steam Cache run on a different IP if you are already hosting another DNS Server or Webserver.
 
Additional note to the Path for the folders: You probably have to configure them in the docker application as a shared folder, but this should work in the newest version if your User has full access to the directories

I know this may sound like the typical questions for less tech-savy people, but: did you check docker with hello world and seriously: did you try turning it off and on again? and at last even if it is reporting as being restarted, just try to lookup any dns from that server by running "ipconfig/flushdns" and then "nslookup google.de 192.168.1.10". And i would stick using the local IPs and not 0.0.0.0 or at least use 127.0.0.1.
 
Oh and by the way what does "docker --version" say?

And it would help if you would post the command you used to start the steamcache-dns

Lets get this command a bit smaller:
docker run --restart unless-stopped -d --name steamcache-dns -p 192.168.1.10:53:53/udp -e UPSTREAM_DNS=1.1.1.1 -e STEAMCACHE_IP=192.168.1.11 -e BLIZZARDCACHE_IP=192.168.1.12 -e FRONTIERCACHE_IP=192.168.1.13 -e WINDOWSCACHE_IP=192.168.1.14 steamcache/steamcache-dns:latest goes to
 
docker run -f --name steamcache-dns -p 192.168.1.10:53:53/udp -e UPSTREAM_DNS=1.1.1.1 -e STEAMCACHE_IP=192.168.1.11 steamcache/steamcache-dns:latest
 
This should run the container in the foreground, thereby enabling console output. And we just remove the other services for now. Have you checked if you opened port 53 on your firewall?