So, I got a few friends now to start playing Closers, but they seem to have a problem with the installer. They say that when they began to install it, the Installer just closes itself after a while. Does somebody know any reason and solution for that?

That's because probably they only installed the nexon launcher or core files for closers. They have to log in closers through nexon site using internet explorer/chrome(not sure firefox doesn't work for some reason) then click game start. Then the game will patch/download itself from there.

Well, now he has the problem that, no matter if he uses firefox, internet explorer or chrome that, when he clicks on "Game Start" just NOTHING happens.

Reinstall Nexon Game Manager.