Yes.

This group is widely documented as "(self-)homeomorphism group of the Cantor set" (by Stone duality).

It admits, for every prime $p$ and $p$-adic field $K$, and $d\ge 2$, the group $\mathrm{PGL}_d(K)$ as a closed subgroup (viewed as acting on the projective space $\mathbb{P}^{d-1}(K)$ which is homeomorphic to a Cantor space), and this group contains a non-abelian free discrete subgroup.

Yes.

This group is widely documented as "(self-)homeomorphism group of the Cantor set" (by Stone duality).

It admits, for every prime $p$ and $p$-adic field $K$, and $d\ge 2$, the group $\mathrm{PGL}_d(K)$ as a closed subgroup (viewed as acting on the projective space $\mathbb{P}^{d-1}(K)$ which is homeomorphic to a Cantor space), and this group contains a non-abelian free discrete subgroup.

Added: here's a direct "Schottky" construction. Suppose that we have a partition of the Cantor set $X=A_+\sqcup A_-\sqcup B_+\sqcup B_-\sqcup Y$ into nonempty clopen subsets, and self-homeomorphisms $a,b$ of $X$ such that $a^{\pm 1}A_{\mp}^c\subset A_\pm$ and $b^{\pm 1}B_{\mp}^c\subset B_\pm$ (ping-pong condition). Then $(a,b)$ freely generates a discrete free subgroup (discrete because the stabilizer of the clopen subset $Y$ is reduced to $\{1\}$). There are plenty of such pairs, for instance coming from suitable loxodromic isometries of trees acting on the boundary.