  The physics of skyrmions, and in particular the issue of how to isolate and
manipulate them individually, is a subject of major importance nowadays in the
community of magnetism. In this article we present an in-depth extension of a
study on this issue that was recently proposed by some of the authors [H. D.
Rosales, et al. Phys. Rev. Lett. \textbf{130}, 106703 (2023)]. More precisely,
we analyse the competition between skyrmions and a chiral spin liquid in a
model on the kagome lattice. We first present an analytical overview of the
low-energy states using the Luttinger-Tisza approximation. We then study the
effect of thermal fluctuations thanks to large-scale Monte-Carlo simulations,
and explore the entire parameter space with a magnetic field $B$, in-plane
$D^{xy}$ and out-of-plane $D^z$ Dzyaloshinskii-Moriya interactions. While
skyrmions and the chiral spin liquid live in different regions of the parameter
space, we show how to bring them together, stabilizing a skyrmion fluid in
between; a region where the density of well-defined skyrmions can be tuned
before obtaining an ordered phase. We investigate in particular the melting of
the skyrmion solid. Our analysis also brings to light a long-range ordered
phase with Z$_3$ symmetry. At last, we initiate the study of this rich magnetic
background on conduction electrons that are coupled to the local spins. We
study how the different chiral magnetic textures stabilized in this model
(skyrmion solid, liquid and gas and chiral spin liquid) induce a topological
Quantum Hall effect. We observe in the ordered skyrmion phase the appearance of
Landau levels which persist even in the skyrmion-liquid regime and gradually
disappear as the skyrmion density decreases to form a gas.
