Login grayed out, logins rejected
Resolved! So, the console clearly showed which files were broken. 

I've downloaded those files again from production server, and uploaded on my test server. Login succeed without any problems now!

I guess it has something to do with transfer mode (auto, binary, text etc) when transferring files via FTP. Transfer mode should be on auto, otherwise files might get corrupted which can cause all kind of (weird) problems. Thanks to Brad for help!