do you need both ethernet connections for mac mini AND imac?

if you connect by wifi you don't need ethernet connections for any of them

if you do not connect by wifi you need ethernet connection for them both if they are to communicate or can't access the internet by other means

if you use ethernet on 1 of them and ask if an ethernet connected device can communicate with a wifi connected device then that depend of if your router is set to bridge the 2 network, same thing with 2.4Ghz wifi and 5Ghz wifi really.

You can run without a router by connecting one Mac to the internet with a cable and sharing the network by WiFi to the second Mac.

Share the internet connection on Mac with other network users - Apple Support

Not the best arrangement because both Macs have to be running for the second to acess the internet.