Hello and welcome to the forum. Indeed not your normal problem.
Can you boot your machine into safemode with networking by tapping the F8 key during POST and see if the problem persists.
Also can you test with no other network node plugged in or otherwise on the network?
Are all network nodes receiving DHCP addresses from the same source (your router) or are one or two manually assigned static values? (this would include any devices including network printers or other network appliances like additional routers)
You say "The modem/router shuts down" can you explain what you mean by this, surely you don't mean it literally as in no power?
Finally can you provide the results of opening a command prompt and typing
ipconfig /all
hit enter
paste the results into your next post?
Thanks
Randy