What is the most powerful result about the maximum of i.i.d. Gaussians? The most used in practice? Given $X_1, \ldots, X_n, \ldots \sim \mathscr{N}(0,1)$ i.i.d., consider the random variables
$$ Z_n := \max_{1 \le i \le n} X_i\,. $$

Question: What is the most "important" result about these random variables?
To clarify "importance", which result has the most other such results as a logical consequence? Which of the results is used most often in practice?

More specifically, it seems to be folklore knowledge among (theoretical) statisticians that the $Z_n$ are "basically the same as" $\sqrt{2 \log n}$, at least asymptotically. (See this related question.)
However, there are many related results of this type, and it seems to be the case that most aren't equivalent, nor imply each other. For example$^*$,
$$ \frac{Z_n}{\sqrt{2 \log n}} \overset{a.s.}{\to} 1 \,, \tag{1}$$
which if nothing else also implies the corresponding results in probability and distribution.
However, it doesn't even imply seemingly also related results (see this other question), like
$$ \lim_{n \to \infty} \frac{\mathbb{E}Z_n}{\sqrt{2 \log n}} =1 \,, \tag{2}$$
(this is exercise 2.17 on p. 49 of $\dagger$), or another folklore result:
$$ \mathbb{E}Z_n = \sqrt{2 \log n} + \Theta(1) \,. \tag{3}$$
Non-asymptotically, it is also known that for each $n$ (see here for a proof),
$$  \sqrt{c \log n} \le  \mathbb{E}Z_n \le \sqrt{2 \log n} \tag{4} $$
for some small $c$. Similar results can also be shown for $|Z_n|$, since $Z_n$ is heavily right-skewed.
The proof of this last result is much more straightforward than the proofs of the other results. My hope had been that the first asymptotic result would have implied all of the other asymptotic ones, so that I could feel confident focusing all of my time and energy in understanding that result. But, again, that seemingly is not true, so now it is unclear to me which I should focus on.
$^*$See pp. 265-267 of the second edition of Galambos, The Asymptotic Theory of Extreme Order Statistics, printed in 1987. It is probably also stated somewhere in the first edition.
$\dagger$ Boucheron, Lugosi, Massart, Concentration Inequalities: A Nonasymptotic Theory of Independence. Aside: This book actually cites Galambos for the result in question, but I can't find it mentioned anywhere in Galambos -- only the first result I mentioned.
 A: In any probabilistic application, the most fundamental object is the distribution, with the moments and limiting properties being derivable from this.  Hence, the most "important" result, in the sense you've described, is the full distribution function $F_{Z_n}(z) = \Phi^n(z)$ (equivalently, the corresponding density function).  In practice, this distributional result is perhaps less illuminating than some of the more basic asymptotic properties you've already listed.  Although it logically implies these asymptotic results, in my view, those results are likely to be more illuminating in understanding the changing nature of the extreme value as we change $n$.
It is clear from your question that you have a good understanding of the extreme value properties in the case of a maximum of IID standard normal random variables.  These properties are all logically derivable from the distribution function for $Z_n$, so that is the most fundamental object at work in this problem.  As in many cases, the most fundamental object is not necessarily the most illuminating, and so you will probably find that you have to make do with knowing all the results, and knowing that they illuminate different aspects of the problem.
