Is it appropriate to mention weird/exciting results in math (or use as cautionary tales why one cannot apply mathematics naively) in say high school level?
Examples of these results include the sphere eversion which turned into a meme (yes, I'm aware it's a true result). Banach-Tarski paradox, Vitali Sets, "1+2+3+4 = $-\frac{1}{12}$" and Hilbert's Hotel.
Additionally, I feel as though $e^{i\pi} = -1$ demonstrates that math despite seemingly random ties up at this beautiful equation.
I would be careful with the type of result for which one needs a lot of new math to digest the explanation.
For example, I would avoid talking about $ 1 + 2+3+.. = -1/12$ because there is basically no way to explain to highschoolers in what sense this could be true. First, one would need a good understanding of the limit and the value of a series (i.e., that this value is not the result of repeated addition) and then one can appreciate the weirdness even more (but the paradox is not removed). I guess zeta function regularization is out of reach for most high schoolers. Many of them will just remember this as one more data point of "math is weird and I can't make sense of it".
You could, however talk about results like the Goldbach conjecture - extremely simple to state and only partly solved recently.
Hilbert's hotel is nice but you would have to explain the notion "infinity" in that context. By the way, there is an upcoming book"Life on the infinite farm" which tells a story about infinity which is more for elementary school. One part is about a cow with an infinite number of legs who wears a show on each foot and gets a new pair of shoes - what should she do? I think that this story about infinity would go very well with kids.
If it's the right amount of "weird", then YES (see Zone of Proximal Development). For example, I often try to show students how $0.\bar{3} = \frac{1}{3}$ implies $0.\bar{9} = 1$. This example alone has so much to unpack, that I often come back to it again and again while teaching rational numbers, series, algorithms, etc.
I could see some mathematical results being so far beyond the grasp of a student that it would at best have no impact, or at worst, be completely de-motivating. It truly is an art to read your audience and pick something that is just beyond their ability, but within their potential.
At any level, I'd ask myself the question "how many new things does the student need to know in order to understand this even at a surface level?". If that number gets beyond 2, I would think that it is a definite no. (So anything involving measure theory is right out.)
The $\sum_{n=1}^{\infty} n = -\frac{1}{12}$ result is going to require zeta functions, analytic continuation, and something about convergent/divergent sums at a minimum. Even if you do this, they are likely to get some nonsense in their head about "wrapping around infinity". I speak from experience here; I was teaching a "Great Ideas in Mathematics" class when a Numberphile video about this was making the rounds on the internet and it seemed timely to talk about it.
Now, $e^{\pi i} = -1$ could maybe work for Calc AB students once they know the chain rule and the derivatives of $\ln,\sin,$ and $\cos$. There, you just need that you should treat $i$ like any other constant and that $e^{i\theta}=\cos\theta+i\sin\theta$, which you can demonstrate by taking a derivative. That is, $$\begin{eqnarray*} \frac{d}{d\theta}\ln\left( \cos\theta + i \sin\theta \right) & = & \frac{\frac{d}{d\theta}\left(\cos\theta + i \sin\theta\right)}{\cos\theta + i \sin\theta} \\ & = & \frac{-\sin\theta + i \cos\theta}{\cos\theta + i \sin\theta} \\ & = & \frac{i\left(\cos\theta + i \sin\theta\right)}{\cos\theta + i \sin\theta} \\ & = & i \end{eqnarray*}$$ So, for some constant $C$, $$ \ln\left( \cos\theta + i \sin\theta \right) = i\theta + C $$ The rest follows by pre-calc methods.
One thing that you didn't mention, that might be worthwhile talking about, is the Weierstrass function: https://en.wikipedia.org/wiki/Weierstrass_function It was designed to combat the misconception that all functions are differentiable except at isolated points and plays an important role in the history of calculus.
As a learner: weird results were one of the things that made me want to learn more. But they mostly had to be weird results of what I already knew, or of something well enough explained for me to follow it. The "birthday paradox" was one example, as I remember, when first learning about probability. I think I may have got that from a Martin Gardner book though, not from what I was learning at school. (I think there were some much better examples, but I'm struggling to remember them at the moment.)
There's no point in showing people something weird to illustrate how their knowledge is valid and yet their expectations are not, if they don't have the knowledge in the first place. Anything that requires university-level topics should immediately be discarded.
A straightforward example that can be approached in a number of ways is the Monty Hall question, should you change your decision once the goat is revealed? Along the same lines, the birthday paradox and its explanation are both easy to communicate and to understand, and the resulting confusion much more easy to work around.
I would bring up some of these truly weird results to bring the humanity into the process. We often teach mathematics as if it has been as it is forever, never changing. These weird results are typically surrounded by a great deal of insight wrought by humans.
For example, $e^{\pi i} = -1$ was not always true. Indeed, its validity stems around the work of Euler. It cemented the meaning of logarithms. Some people at the time were running on the belief that $log(-x) = log(x)$, which was a natural way of thinking if you didn't want to have complex numbers.
In fact, tying his work into Analytic Continuation would be a way to show the kind of way mathematicians think. There's countless ways one could have gone about extending logarithms, but this was the one way which satisfied the expectations of mathematicians.
Likewise I find there's no point in teaching the Banach-Tarski paradox unless you want to get into the fact that there is not a universal agreement about how to axiomize set theory surrounding the axiom of choice. Few outside of the true set theorists will ever care about whether the axiom of choice is true or not. But the fact that mathematicians are divided on it is meaningful. It could also be used to show how some theories may start out being proven in ZFC, and only later do proof in ZF surface.
Indeed, without that, the only conclusion you can really take from Banach-Tarski is that mathematicians are idiots, because only an idiot would fail to see that you can't make one sphere into two of the same size just by cutting rotating and gluing! You have to show enough to demonstrate why it really bothers mathematicians that they can't just blindly accept the axiom of choice, as intuitive as it is.
Show that humans are still figuring out how mathematics should work.
