To test it by pinging to google.nl and play a game at the same time. When the game hangs I also see "ping: sendto: No route to host - Request timeout for icmp_seq". Steam also reconnects at those moments. It looks like it can't handle all the packets it receives and sends and just drops the connection for 1 second sometimes.
Sorry, but that's nonsense. In case there is a driver related problem you should post kernel logs showing the issue. At first glance, this could also be related to Energy Efficient Ethernet as it might sometimes causes the link to be dropped when there is no activity.