Ok, I think I know what you're saying
No, you cannot get a virus when you computer is turned off, however, your network is going through LAN and the BIOS have an option called WOL (Wake On LAN). If this is enabled, something like a network ping could boot your computer up. So really, your internet is always on, but it's not always communicating back and forth with your computer.
And yes, update it! Some people think updating doesn't do much, but really it optimizes certain functions, secures some holes, and adds features, so you should try and update it as much as possible.
And also yes, it is possible for a computer to spread viruses to another one on the network, but only if the virus instructs the computer to do so (most viruses aren't this complex though).
If you're concerned with viruses, I strongly suggest you get a good
antivirus, as it's an investment for your computer's health and reliability. Try and pickup the latest version (I think 2006) of Norton AntiVirus. In my opinion, it's the best on the market