  In this paper, we study several finite approximation properties of
topological full groups of group actions on the Cantor set such that free
points are dense. Firstly, we establish that for such a distal action $\alpha$
of a countable discrete group $G$ on the Cantor set, the topological full group
$[[\alpha]]$ is amenable if and only if $G$ is amenable. This result is
obtained through a novel method that detects hyperfiniteness in certain sofic
approximation graph sequences of finitely generated subgroups of $[[\alpha]]$.
We also provide estimates for related F{\o}lner functions. Next, we obtain
negative results on the amenability of topological full groups for actions with
zero topological entropy by calculating the topological entropy of certain
examples provided by Elek and Monod. Furthermore, we demonstrate that the
topological full group $[[\alpha]]$ of a minimal topologically free residually
finite action $\alpha$ on the Cantor set is locally embeddable in the class of
finite groups (LEF). This generalizes a result previously obtained by
Grigorchuk and Medynets in the case of minimal $\mathbb{Z}$-actions. As an
application, we show that topological full groups of certain Toeplitz subshifts
on free groups are LEF and therefore sofic.
