Btukfyl

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
$e^{ipi} = -1$ 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.

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 $e^{itheta}=costheta+isintheta$, which you can demonstrate by taking a derivative. That is,
$$begin{eqnarray*}
frac{d}{dtheta}lnleft( costheta + i sintheta right)
& = & frac{frac{d}{dtheta}left(costheta + i sinthetaright)}{costheta + i sintheta} \
& = & frac{-sintheta + i costheta}{costheta + i sintheta} \
& = & frac{ileft(costheta + i sinthetaright)}{costheta + i sintheta} \
& = & i
end{eqnarray*}$$
So, for some constant $C$,
$$ lnleft( costheta + i sintheta right) = itheta + C $$
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.

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.