Maximum PC

I've asked this before, but never got around to seeing the results (so sorry if this is considered a double post)...

I have two computers that use the same router. They are not always plugged in at the same time. So, what happens if one gets a virus, but the other computer is not plugged in. Is it possible for the router to "house" this virus so that when the other computer is plugged in (and the first computer is subsequently unplugged from the router so they are never plugged in at the same time)?

Is it possible for a virus to hide in the flash area of a router (because I've heard of people having issues where a router would spam them or something similar, denial of service, etc).

In all, is it possible for a router to become infected with a virus, trojan, worm, etc? How about other components, like a keyboard (I want to say no since a keyboard and mouse do not have any type of internal data storage device, but just want to be sure).

Yes to all of the above.

A router has an OS, of sorts, and can house a virus and pass it along.

Whether there are virii in the wild that can do this, I don't know .. but it is possible