Is Medal of Honor: Cooperative free?

Yes. Medal of Honor: Cooperative is released under Free fan mod (see the ModDB page for terms) and costs nothing. Neither it nor Medal of Honor: Allied Assault War Chest has any paid content.

What do I need to run Medal of Honor: Cooperative?

Medal of Honor: Allied Assault War Chest, which is itself free. Medal of Honor: Cooperative is an add-on and will not run standalone.

How do I install Medal of Honor: Cooperative?

The PlayBound Launcher installs it in one click from Medal of Honor: Allied Assault War Chest's mods list. You can also install it by hand — full steps are on this page.