Under system preferences, security & privacy, firewall - try disabling your firewall temporarily as a test. Is this the first time you’ve installed Roon or was it working before on another device in your house?
I’m not sure if your Mac-mini is connected via wifi or ethernet cable, and I assume by ‘green lock light’ you mean the ‘connected’ light on the Apple-system-preferences-network panel. Two things you can check - 1) if you remember the IP address of your previous Roon-core device, make sure it is the same as your new Mac-mini. or 2) login to your Mac-mini, open a terminal screen, and ping the IP address of your DS (which you can get from the DS’s display screen.). The ping should go directly from the Mac-mini to the DS and not go through a router/default-gateway.
Your MAC is 192.168.1.4 and received this IP via DHCP. The default-gw for your network is 192.168.1.1. This is all good. The weird thing is you said your DS has an IP of 192.168.1.1 - this is the same IP that your MAC thinks should be the default-gw. The DS should have an IP of something like 192.168.1.5 with a default-gw of 192.168.1.1.
If you turn your DAC off with the switch on the back, wait for a couple of minutes, and turn it back on, it should ask (automatically) for a new IP address. After it restarts, wait a few minutes and check the IP address. If it is unchanged and/or Roon can’t find it, post a picture of your DS screen that shows the IP addresses here.