Extreme high-frequency-peaked BL Lac objects (EHBLs) are blazars that exhibit extremely energetic synchrotron emission. They also feature nonthermal gamma-ray emission whose peak lies in the very high-energy (VHE, E > 100 GeV) range, and in some sources exceeds 1 TeV: this is the case for hard-TeV EHBLs such as 1ES 0229+200. With the aim of increasing the EHBL population, 10 targets were observed with the MAGIC telescopes from 2010 to 2017, for a total of 265 hr of good-quality data. The data were complemented by coordinated Swift observations. The X-ray data analysis confirms that all but two sources are EHBLs. The sources show only a modest variability and a harder-when-brighter behavior, typical for this class of objects. At VHE gamma-rays, three new sources were detected and a hint of a signal was found for another new source. In each case, the intrinsic spectrum is compatible with the hypothesis of a hard-TeV nature of these EHBLs. The broadband spectral energy distributions (SEDs) of all sources are built and modeled in the framework of a single-zone, purely leptonic model. The VHE gamma-ray-detected sources were also interpreted with a spine─layer model and a proton synchrotron model. The three models provide a good description of the SEDs. However, the resulting parameters differ substantially in the three scenarios, in particular the magnetization parameter. This work presents the first mini catalog of VHE gamma-ray and multiwavelength observations of EHBLs.