  Let $G$ be a discrete countable infinite group. We show that each topological
$(C,F)$-action $T$ of $G$ on a locally compact non-compact Cantor set is a free
minimal amenable action admitting a unique up to scaling non-zero invariant
Radon measure (answer to a question by Kellerhals, Monod and R{\o}rdam). We
find necessary and sufficient conditions under which two such actions are
topologically conjugate in terms of the underlying $(C,F)$-parameters. If $G$
is linearly ordered Abelian then the topological centralizer of $T$ is trivial.
If $G$ is monotileable and amenable, denote by ${\cal A}_G$ the set of all
probability preserving actions of $G$ on the unit interval with Lebesgue
measure and endow it with the natural topology. We show that the set of
$(C,F)$-parameters of all $(C,F)$-actions of $G$ furnished with a suitable
topology is a model for ${\cal A}_G$ in the sense of Forman, Rudolph and Weiss.
If $T$ is a rank-one transformation with bounded sequences of cuts and spacer
maps then we found simple necessary and sufficient conditions on the related
$(C,F)$-parameters under which (i) $T$ is rigid, (ii) $T$ is totally ergodic.
It is found an alternative proof of Ryzhikov's theorem that if $T$ is totally
ergodic and non-rigid rank-one map with bounded parameters then $T$ has MSJ. We
also give a more general version of the criterium (by Gao and Hill) for
isomorphism and disjointness of two commensurate non-rigid totally ergodic
rank-one maps with bounded parameters. It is shown that the rank-one
transformations with bounded parameters and no spacers over the last subtowers
is a proper subclass of the rank-one transformations with bounded parameters.
