What are the main responsibilities for work in small, mid or enterprise size companies in Network Department?

To check the connections for all devices and make sure that it's working, also able to give all devices a proper IP and to use the matched linking devices for it, also to maintain any cable problems or even to connect all cables and routers together and to make sure every networks are in the same range, also to connect the branches together and make sure that you have a good firewall for all of your connections.

visibility visibility visibility 

first thing that should be done scan your network, u shoId be familiar with each device connected.

segmenting, break down the bigger network into small easy manageable networks.

keep an eye on the gateways, they make u have a good control over the networks, and are the first point of  tshooting.

NTP and SysLog servers, they will save u alot of time; I mean ALOT.

monitor traffic, u will be surprised with the amount of infos u are getting by monitoring; misconfigured devices, infected/compromised devices, and network/user abuse, these  will help u into finding network flows and better give u a clear vision into enhancing  both efficiency and security.