My unpatched PC running Windows is connected to a network which contains other "infected" computers, can I get infected by viruses without any interaction on my part?
All the other infected PC(s) have to do is find a vulnerability in an unpatched application visible on the network on your PC and exploit it with vulnerabilities they know about.
This doesn't just apply to Windows either. Any OS that is network facing and has exploitable vulnerabilities can be attacked this way.
Yes you can, these type of virusses are known as Worm virusses and may try to exploit vulnerabilities in services you are running which are network facing.
Sure... if some of the services running on the pc are vulnerable, it's quite possible a virus could exploit this vulnerability and infect it.