Given that all machines lose their IP address at the same time, the router appears to be the problem but it's worth disconnecting all but one machine and seeing if the same problems occur.
Monitor the remaining NIC with Wireshark - if you issue an ipconfig /renew command you should see 255.255.255.255 Discovery broadcasts from the client looking for a DHCP server, then a unicast back from the server with a DHCP Offer (the first two of the 5 message types used in DHCP). That would be a starting point. If things seem to improve, add another machine and keep monitoring the DHCP messages. Ideally you want it to break while you're monitoring to see what's actually happening - if things improve with only one machine connected, start adding more.
Worth checking the DHCP Client service is running on all machines too.