Hi and welcome, you aren't an it guy but your explanation is perfect, detailed! Well done.
Ok first of all we need to understand if all computers connected to domain are really equally configured, these PC have a static ip or is assigned by DHCP? If so, assign a static ip.
when you lose connection, can you ping the server? And the others PC? Actually your connection is not lost because you can easily reconnect with a logout and login. If you haven't a connection you can't connect to your domain. Are there any errors on the clients side in event viewer?