As part of the T-REX project, a number of R&D and prototyping activities have been carried out during the last years to explore the applicability of Micromegas-read gaseous TPCs in rare event searches like double beta decay (DBD), axion research and low-mass WIMP searches. While in the companion paper we focus on DBD, in this paper we focus on the results regarding the search for dark matter candidates, both axions and WIMPs. Small ultra-low background Micromegas detectors are used to image the x-ray signal expected in axion helioscopes like CAST at CERN. Background levels as low as $0.8times 10^{-6}$ c keV$^{-1}$cm$^{-2}$s$^{-1}$ have already been achieved in CAST while values down to $sim10^{-7}$ c keV$^{-1}$cm$^{-2}$s$^{-1}$ have been obtained in a test bench placed underground in the Laboratorio Subterraneo de Canfranc. Prospects to consolidate and further reduce these values down to $sim10^{-8}$ c keV$^{-1}$cm$^{-2}$s$^{-1}$will be described. Such detectors, placed at the focal point of x-ray telescopes in the future IAXO experiment, would allow for 10$^5$ better signal-to-noise ratio than CAST, and search for solar axions with $g_{agamma}$ down to few 10$^{12}$ GeV$^{-1}$, well into unexplored axion parameter space. In addition, a scaled-up version of these TPCs, properly shielded and placed underground, can be competitive in the search for low-mass WIMPs. The TREX-DM prototype, with $sim$0.300 kg of Ar at 10 bar, or alternatively $sim$0.160 kg of Ne at 10 bar, and energy threshold well below 1 keV, has been built to test this concept. We will describe the main technical solutions developed, as well as the results from the commissioning phase on surface. The anticipated sensitivity of this technique might reach $sim10^{-44}$ cm$^2$ for low mass ($<10$ GeV) WIMPs, well beyond current experimental limits in this mass range.
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