Foundational results dependent on/equivalent to the continuum hypothesis or its negation?

As it is stated in the comments, one reference is Sierpinski's book, Hypothese Du Continu, though it is not in English.

Another reference is Propositions Equivalent to the Continuum Hypothesis.

See also Some propositions equivalent to the continuum hypothesis and The continuum hypothesis (CH) and its equivalent.

You may be also interested in Eliminating the Continuum Hypothesis.

Let me also state one equivalent of CH. I have taken it from Interactions between (set theory, model theory) and (algebraic geometry, algebraic number theory ,...):

Let $R$ be a ring and $D(R)$ its unbounded derived category. Let $D^c(R)$ be the full subcategory of compact objects (in the explicit example below it is spanned by bounded complexes of f.g. projective modules). We say that $D(R)$ satisfies Adams representability if any cohomological functor $D^c(R)^{op}\rightarrow Ab$, i.e. additive and taking exact triangles to exact sequences, is isomorphic to the restriction of a representable functor in $D(R)$ (in particular it extends to the whole $D(R)$), and any natural transformation between restrictions of representable functors $D^c(R)^{op}\rightarrow Ab$ is induced by a morphism in $D(R)$ between the representatives.

Let $\mathbb C\langle x,y\rangle$ be the ring of noncommutative polinomials on two variables. The statement '$D(\mathbb C\langle x,y\rangle)$ satisfies Adams representability' is equivalent to the continuum hypothesis.

For another interesting equivalent of CH see: Reductions between certain incidence problems and the continuum hypothesis.

Concerning your first question, there is a simple, if not "self-evident", order-theoretic statement equivalent to $CH$ admitting a generalization equivalent to $GCH$:

• If $L$ is a linear ordering of size $2^{\omega}$, then $L$ embeds every cardinal less than $2^{\omega}$ or $L^*$ ($L$ reversed) embeds every cardinal less than $2^{\omega}$.

This statement can be read in general terms as follows: In order to arrange $2^{\omega}$ points in a line one cannot bypass a smaller cardinal (in the sense that it will appear directly or reversed).

The following generalization is equivalent to $GCH$:

• For every cardinal $\lambda$, if $L$ is a linear ordering of size $\lambda$, then $L$ embeds every cardinal less than $\lambda$ or $L^*$ embeds every cardinal less than $\lambda$.

It says, in general terms, that for every cardinal $\lambda$, in order to arrange $\lambda$ points in a line one cannot bypass a smaller cardinal.

It is important to remark that the regularity thus stated is trivially true in the finite realm (for finite $\lambda$), contrary to $GCH$ (there are, in general, many numbers between $n$ and $2^n$). Therefore, it is, at least, a more uniform statement generalizing a fact of the finite realm for which, presumably, we have a more reliable "intuition". This feature is present in the usual axioms of set theory.

Addendum:

I have just recalled the paper below which seems to be relevant to your question. It connects $CH$ with a more or less concrete problem of machine learning:

• Shai Ben-David, Pavel Hrubeš, Shay Moran, Amir Shpilka and Amir Yehudayoff, Learnability can be undecidable, Nat. Mach. Intell. 1 (2019) 44–48, doi:10.1038/s42256-018-0002-3.

($CH$ is equivalent to a version of learnability)