This is somewhat of a non-answer to the second question (I do need a little bit of AC) so I apologize. But I thought it might be worthwhile for its generality.

The **uncountable example** I had in mind is $\beta \mathbb{N}$, the set of ultrafilters on the natural numbers, i.e., the Stone–Čech compactification of the discrete space $\mathbb{N}$.

- $(\beta \mathbb{N},+)$ is a semigroup (with + extending the usual addition on $\mathbb{N}$ to $\beta \mathbb{N}$ in such a way that addition with a fixed right hand side is continuous)
- The shift $s(p)= 1+ p$ makes $(\beta \mathbb{N},s)$ a dynamical system (because addition with natural numbers is left-continuous, too)
- Its minimal systems are exactly the minimal left ideals of the semigroup
- Its cardinality is $2^{2^{\aleph_0}}$ (i.e., the size of the power set of the reals)
- The minimal left ideals are universal minimal systems for discrete time, so one minimal left ideal induces minimal systems everywhere

To prove all this, if I'm not mistaken, you require "only"

- The ultrafilter lemma for (the power set of) $\mathbb{N}$ so that you actually get the space (the ultrafilter lemma is strictly weaker than AC)
- An application Zorn's Lemma to find a minimal left ideal

So that's not "a lot" of AC to get one minimal system everywhere (which is why I thought it'd be worthwhile).

On the other hand, if you assume, e.g., ZF+AD you do not find any free ultrafilters on $\mathbb{N}$, so no $\beta \mathbb{N}$, and I have no idea what the dynamics look like then...

metricspace without the axiom of choice, the idea being to use the Hausdorff metric on compact subsets. (The remark is credited to S. Simpson). $\endgroup$