Is the weakSelf/strongSelf dance really necessary when referencing self inside a non-retained completion called from a UIViewController?

As I believe you correctly diagnosed, using self will not necessarily cause strong reference cycle in this scenario. But this will retain the view controller while the network operation completes, and in this case (as in most cases), there's no need to. Thus, it may not be necessary to do use weakSelf, but probably prudent to do so. It minimizes the chance of an accidental strong reference cycle and leads to more efficient use of memory (releasing the memory associated with the view controller as soon as that view controller is dismissed rather than unnecessarily retaining the view controller until after the network operation is complete).

There is no need for the strongSelf construct, though. You can:

- (void)makeAsyncNetworkCall
{
    __weak typeof(self) weakSelf = self;
    [self.networkService performAsyncNetworkCallWithCompletion:^{
        dispatch_async(dispatch_get_main_queue(), ^{
            [weakSelf.activityIndicatorView stopAnimating];
        });
    }];
}

You only need the weakSelf/strongSelf combination where it's critical to have a strong reference (e.g., you're dereferencing ivars) or if you need to worry about race conditions. That does not appear to be the case here.

I think the issue is that the networkService may keep a strong reference to the block. And the view controller may have a strong reference to the networkService. So the possible cycle of VC->NetworkService->block->VC could exist. However, in this case, it's usually safe to assume that the block will be released after it has run, in which case the cycle is broken. So, in this case, it isn't necessary.

Where it is necessary is if the block is not released. Say, instead of having a block that runs once after a network call, you have a block that is used as a callback. i.e. the networkService object maintains a strong reference to the block and uses it for all callbacks. In this case, the block will have a strong reference to the VC, and this will create a strong cycle, so a weak reference is preferred.

No, If your self.networkService don't use it as a block property you should be fine