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 gaseous Time Projection Chambers (TPCs) with Micromesh Gas Structures (Micromegas) in rare event searches like double beta decay, axion research and low-mass WIMP searches. While in the companion paper we focus on double beta decay, in this paper we focus on the results regarding the search for dark matter candidates, both axions and WIMPs. Small (few cm wide) ultra-low background Micromegas detectors are used to image the axion-induced x-ray signal expected in axion helioscopes like the CERN Axion Solar Telescope (CAST) experiment. Background levels as low as 0.8 × 10−6 counts keV−1 cm−2 s−1 have already been achieved in CAST while values down to ∼10−7 counts keV−1 cm−2 s−1 have been obtained in a test bench placed underground in the Laboratorio Subterráneo de Canfranc (LSC). Prospects to consolidate and further reduce these values down to ∼10−8 counts keV−1 cm−2 s−1 will be described. Such detectors, placed at the focal point of x-ray telescopes in the future International Axion Observatory (IAXO), would allow for 105 better signal-to-noise ratio than CAST, and search for solar axions with gaγ down to few 1012 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 ∼ 0.300 kg of Ar at 10 bar, or alternatively ∼ 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 ∼10−44 cm2 for low mass (<10 GeV) WIMPs, well beyond current experimental limits in this mass range.