You can use a null-modem cable; alternately, you can use a USB or FireWire cable.
Two computers
a cord
You can buy a firewire cord. That is the fastest and easiest way to connect two computers.
wireless
An HDMI cable is not designed to connect two computers together.
Cross cable.
linksys
yes
You can connect them with an ethernet cable, or by WiFi if they have wireless cards.
Buy a lan router, they can be used with lan cable or wireless to connect all your computers through a network. If there are only two computers that you wish to connect then you only need on lan cable which you connect in the back of each of those two computers to create a network.
The safest and easiest way to connect computers across time is with a VPN. A VPN is a virtual private network that makes a tunnel between two or more computers.
I believe that it is possible to connect to computers to one monitor. I have worked with brand new LCD monitors with two inputs - one DVI and one VGA. The monitor auto-senses which port is being used. So in theory, use the auto detect button on the front of the LCD panel to switch between computers.