How do I add a dark screen that covers all other views (including navigation bar and status bar)?

You have to do this:

CGRect screenRect = [[UIScreen mainScreen] bounds];
_darkCoverView = [[UIView alloc] initWithFrame:screenRect];
_darkCoverView.backgroundColor = [UIColor blackColor];
_darkCoverView.alpha = 0.5;

[[[[UIApplication sharedApplication] delegate] window] addSubview:_darkCoverView];

It works for me, and the status bar is covered as well :)

You can try this method

    UIWindow* window = [[UIWindow alloc] initWithFrame:[[UIScreen mainScreen] bounds]];
    window.windowLevel = UIWindowLevelAlert;
    window.opaque = NO;

    [window addSubview:someView];

    // window has to be un-hidden on the main thread
    [window makeKeyAndVisible];

Where "view" - is your custom popup.