We have a computer in one room connected by cable and a wi-fi modem. Can we move the screen to an other room and leave the modem and computer where it is?
What you will need are some really long video, keyboard and mouse cables. You can then run these from where your computer is now to where you want the screen to be.
There is a limit on how long these cables can be (though I'm not sure what it is at the moment - I'll check) but you might be able to get wireless repeaters. However (as @Darakthar points out) that would mean running three cables between rooms and you won't get very far. There are ways of using a single cable but require hardware at either end.
You say you have a wi-fi modem. What sort? Does the computer connect to that via wireless, ethernet or USB connection? If it's either of the first two then you can move the whole computer but leave the modem where it is. Just get a wireless dongle (if you don't already have one) for the computer.
Technically Yes. But it's likely cost prohibitive.
You could get a Wireless KVM solution, or a KVM extender that would allow you to combine the necessary cables into a single cat5e network cable. But these devices are not usually inexpensive.
Also, doing so could have an impact on image quality on the monitor as well as video performance.
You'd probably be better off buying a laptop and using that instead.
Of course! Where are many way to do it:
First variant is a joke :)