You can certainly have an infinite family of mutually almost orthogonal functions: for every $n$, let $f_n$ be the sequence of $n$ $1$s, followed by $n$ $-1$s, then $n$ $1$s, then $n$ $-1$s, etc. But can there be an uncountable family? . . .

@abeaumont: Some of the results in our paper could be interpreted that way. For example, you can put a (not group action invariant) measure on $S_\infty$ such that every null $S$ admits an $S$-conditionally convergent series. We define a notion of "jumbling" such that every non-jumbling set $S$ admits an $S$-conditionally convergent series. My guess would be that there are other things you could say along these lines. Please post an update to your question at some point if you end up finding any interesting ones!

@NotMike: Good observation. The property of $\omega^*$ mentioned in YCor's comment is, in general, equivalent to the algebra of regular open sets being countably closed, which is a little stronger than being $\omega_1$-distributive.

@HenrikRüping: Every countable dense subspace of the Cantor space is homeomorphic to $\mathbb Q$. So the Cantor space is a metrizable compactification of $\mathbb Q$.

The general idea is that if you see a property of $\beta X$ that can be expressed in a sufficiently "first-order" way, then you can reflect that property down to a metrizable compactification of $X$. You can see more about this kind of thing if you look at/around Lemma 3.2 here: wrbrian.wordpress.com/wp-content/uploads/2012/01/….

Nice question! I don't have time to check the details at the moment, but here is an idea that I think should work. First, prove the Stone-Cech compactification $\beta X$ has all the properties you want except for metrizability. Next, you can find a "metrizable reflection" of $\beta X$; it has many of the same properties as $\beta X$, and in particular this property of not giving new paths should reflect down. A metrizable reflection of $\beta X$ is obtained by taking the (countable!) lattice of open sets from $\beta X$ that are in a countable elementary submodel, then spacifying that lattice.

If you look at the "natural addition" on ordinals (en.wikipedia.org/wiki/Ordinal_arithmetic#Natural_operations), then you can prove a version of van der Waerden's Theorem for any cardinal, or in fact any indecomposable ordinal. And you can prove it by copying over the ultrafilters-based proof for $\omega$, making the appropriate changes. But I'm not sure this is really in the spirit of your question . . . to me, it feels like it has little to do with ordinals or cardinals at all. It's more a theorem about arbitrary commutative semigroups (known as the Gallai Theorem).