We can have many handlers: touches handler, UIControl handler (buttons, sliders), performSelector, CADisplayLink, NSTimer events, Gesture Recognizer, accelerometer handler, and UIView animation completion block, and some other ones.
Are all of them in the same thread? That is, only one of them can be running at the same time?
Can some other method or handler be part of another thread and therefore can create race conditions?
In general, you'll find that most simple applications on iOS tend to perform almost every action on the main thread. As you noted, the instant that you bring multithreading into the picture you add another set of tricky issues to watch out for. Many developers don't want to bother with this added complexity, or are unfamiliar with GCD or threading in general, so they avoid doing anything on a background thread or GCD queue.
Several of the items you list in your question involve interactions with UIKit, and in general those interactions must occur on the main thread (iOS 4.x added the ability to perform some drawing functions in the background, though). You receive touch and other related events on the main thread. If you wish to update most aspects of an interface, the safe way to do that is by performing these updates on the main thread.
Timers (NSTimer, CADisplayLink) can have their updates be fired on a background thread by attaching them to an NSRunLoop operating on that background thread. You rarely see people do this, but it can be done. Usually, people configure timers on the main run loop, which causes callbacks to be delivered on the main thread.
When performing animations, the animations themselves will run on a background thread (you see that they don't stop while you're blocking the main thread with something else), but you'll almost always receive a completion block or callback on the main thread when they're done. If I remember correctly, there are one or two exceptions to this and they are noted as such in Apple's documentation. Having these callbacks trigger on the main thread is a safe approach when dealing with developers who might not realize what's going on behind the scenes.
All that said, there are very good reasons to want to add multithreading to your application. Because all user interface updates and touch interactions occur on the main thread, if you have something that is computationally expensive or that simply will take a lot of time to perform, if you run this on your main thread you'll appear to have frozen your application. This is a terrible user experience, so you want to move this task onto a background thread so that the user can keep interacting with your application while this is going on. Additionally, more and more iOS devices are shipping every day with multiple cores in them, and balancing your work load across these cores and being efficient with this hardware requires some degree of concurrent processing.
People have written books about best practices when making code multithreaded, and you can find a lot of questions about this here, so I won't go into too much detail. What I can tell you is that you should read Apple's Concurrency Programming Guide and watch the WWDC videos from the last two years that deal with Grand Central Dispatch. With GCD, Apple has made it a lot easier to add multithreading to your application in an efficient and (relatively) safe manner, and I encourage you to look into this for your own applications.
For example, I have an open source iOS application that performs detailed rendering of molecular structures. I render each frame on a background GCD queue because sometimes they take more than 1/60th of a second to process, and in those cases they'd cause touch events to be dropped and the interface to stutter if this was all on the main thread. Additionally, I've seen up to a 40% performance boost by doing this when running on the newer multicore devices. To avoid race conditions, I wrap interactions with shared data structures and contexts in serial dispatch queues so that only one action can be using a resource at a time, no matter what thread a particular block is running on. This only required the addition of a few lines of code, but the performance and user experience benefits were huge.