How can I make it so if I fade my navigation bar out, then actually hide it programmatically, that if I undo this there's no timing issues?

Why not use the animationCompleted delegate or block?

The way I'd do this is simply create a BOOL flag that I'd call something like isTransitioning, such that once the hiding/unhiding process starts, the hideControls method returns immediately if a transition is in progress. That way you're not messing with touch events; you're directly stopping the unwanted glitches without causing side-effects elsewhere (you'll need to declare isTransitioning as a property/ivar outside of the method, obviously):

- (void)hideControls:(BOOL)hidden {

    //Check there isn't a hide/unhide already in progress:
    if(self.isTransitioning == YES) return;

    //if there wasn't already a transition in progress, set
    //isTransitioning to YES and off we go:
    self.isTransitioning = YES;

    self.navigationController.view.backgroundColor = self.view.backgroundColor;
    int statusBarHeight = [UIApplication sharedApplication].statusBarFrame.size.height;

    [UIView animateWithDuration:0.35 animations:^{
        [[UIApplication sharedApplication] setStatusBarHidden:hidden withAnimation:UIStatusBarAnimationFade];

        if (hidden) {
            self.navigationController.navigationBar.alpha = 0.0;
            self.instructionsLabel.alpha = 0.0;
            self.backFiftyWordsButton.alpha = 0.0;
            self.forwardFiftyWordsButton.alpha = 0.0;
            self.WPMLabel.alpha = 0.0;
            self.timeRemainingLabel.alpha = 0.0;
        }
        else {
            self.navigationController.navigationBar.alpha = 1.0;
            self.instructionsLabel.alpha = 1.0;
            self.backFiftyWordsButton.alpha = 1.0;
            self.forwardFiftyWordsButton.alpha = 1.0;
            self.WPMLabel.alpha = 1.0;
            self.timeRemainingLabel.alpha = 1.0;
        }

        [self.view layoutIfNeeded];
    }];

    // Perform an "actual" hide (more than just alpha changes) after the animation finishes in order to regain that touch area
    if (hidden) {
        double delayInSeconds = 0.35;
        dispatch_time_t popTime = dispatch_time(DISPATCH_TIME_NOW, (int64_t)(delayInSeconds * NSEC_PER_SEC));
        dispatch_after(popTime, dispatch_get_main_queue(), ^(void) {
            [self.navigationController setNavigationBarHidden:YES animated:NO];
            self.textToReadLabelPositionFromTopConstraint.constant = TEXT_LABEL_DISTANCE + self.navigationController.navigationBar.frame.size.height + statusBarHeight;

        //Unset isTransitioning now we're finished:
        self.isTransitioning = NO;
        });
    }
    else {
        [self.navigationController setNavigationBarHidden:NO animated:NO];
        self.textToReadLabelPositionFromTopConstraint.constant = TEXT_LABEL_DISTANCE;

        //Unset isTransitioning now we're finished:
        self.isTransitioning = NO;
    }
}

I haven't directly tested this code, but you can see what I'm getting at, I'm sure.

The other answers are helpful, but one thing you should probably do is instead of hard-coding your animation duration to 0.35, try using UINavigationControllerHideShowBarDuration. This will make your app more resilient to changes to UIKit behavior.

EDIT #2

looking at the docs, hitTest:withEvent: simply calls pointTest:withEvent: on all subviews of the view. It clearly states that views with alpha level less than 0.01 are ignored. I think we are on the right path here, just need to explore further. I'm sure there's a way to have a view with alpha == 0.0f pass through touches to any views beneath it. Hopefully you (or someone else here) will get it. If I have time I'll dive into some code and try to help further.

EDIT #1: try overriding pointInside:withEvent:

Sorry for the stream of consciousness answer here. Normally I would either test this myself or paste code from a production app but I'm too busy right now.

I think overriding pointInside:withEvent will work:

- (BOOL)pointInside:(CGPoint)point withEvent:(UIEvent *)event {
    if (self.alpha == 0.0f) {
        return NO;
    }
    else {
        return [super pointInside:point withEvent:event];
    }
}

ORIGINAL ANSWER: I would try subclassing UINavigationBar and overriding hitTest:withEvent: so that the navigation bar ignores touches while it is invisible. I didn't test this but should be something like:

- (UIView *)hitTest:(CGPoint)point withEvent:(UIEvent *)event
{
    // if alpha == 0.0f, the nav bar should ignore touches
    if (self.alpha == 0.0f) {
        return nil;
    }

    // else hitTest as normal
    else {
        return [super hitTest:point withEvent:event];
    }
}

If testing for alpha == 0.0f isn't how you want to decide to ignore touches, you could also add your own property and do it that way.

@property (nonatomic) BOOL ignoreTouches;

In your hitTest:withEvent: implementation check for if (ignoreTouches) and return nil.

To turn off receiving touch events while you animate the UINavigationBar then subclass it and add the following method:

-(BOOL) canBecomeFirstResponder{
    NSLog(@"Decide if to allow the tap through");
    if (self.hiding) return NO;
    return YES;
}

Then you can control if and when you allow touches to be responded to.

Keep in mind that UINavigationBar is a subclass of UIResponder which allows you to override this and many other methods. I also often forget to look up the inherit chain.

You could use animateWithDuration:animations:completion:. The completion block is executed after the animations have completed (source). It also has an added benefit; if you decide to change the timing of the animations in the future, you won't have to worry changing the timing in two places.

[UIView animateWithDuration:0.35 animations:^{
    [[UIApplication sharedApplication] setStatusBarHidden:hidden withAnimation:UIStatusBarAnimationFade];

    if (hidden) {
        self.navigationController.navigationBar.alpha = 0.0;
        self.instructionsLabel.alpha = 0.0;
        self.backFiftyWordsButton.alpha = 0.0;
        self.forwardFiftyWordsButton.alpha = 0.0;
        self.WPMLabel.alpha = 0.0;
        self.timeRemainingLabel.alpha = 0.0;
    }
    else {
        self.navigationController.navigationBar.alpha = 1.0;
        self.instructionsLabel.alpha = 1.0;
        self.backFiftyWordsButton.alpha = 1.0;
        self.forwardFiftyWordsButton.alpha = 1.0;
        self.WPMLabel.alpha = 1.0;
        self.timeRemainingLabel.alpha = 1.0;
    }

    [self.view layoutIfNeeded];
}
completion:^(BOOL finished){
    // Perform an "actual" hide (more than just alpha changes) after the animation finishes in order to regain that touch area
    if ( finished ) {
        if (hidden) {
            [self.navigationController setNavigationBarHidden:YES animated:NO];
            self.textToReadLabelPositionFromTopConstraint.constant = TEXT_LABEL_DISTANCE + self.navigationController.navigationBar.frame.size.height + statusBarHeight;
        }
        else {
            [self.navigationController setNavigationBarHidden:NO animated:NO];
            self.textToReadLabelPositionFromTopConstraint.constant = TEXT_LABEL_DISTANCE;
        }
    }
}];

Based on the comments, you could disable the navigation bar before the animations start, and re-enable in the completion block, but that is up to you test what works best for you..

Rather than messing with views in the nav-controller stack, why not have a FullScreenViewController (from UIViewController) and make that the root of your app. Then add the NavController (and its stack) on top of that. When the time comes, just fade the whole NavController, exposing your FullScreenViewController.

(You could do this idea upside down, too -- something like this (typed in browser -- lots of syntax errors!):

UIViewController *vc = // ... FullScreenViewController
vc.modalTransitionStyle = UIModalTransitionStyleCrossDissolve;
[navController presentViewController: vc animated: YES completion: nil];

NOTE: you can also make use of childViewControllers to have an abstract class that contains the VC which will be both the full-screen and the not-full screen versions, and then just steal its view, as desired.