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We provide detailed analysis of the curvature structure of the spacetime around regular black holes (RBHs) governed by general relativity (GR) combined with nonlinear electrodynamics (NED), characterized by electric charge Q, and degree of nonlinearity n. We consider special class of the backgrounds introduced by Toshmatov et al. [Phys. Rev. D 98, 028501 (2018)] that have proper Maxwell weak-field limit of the NED sector. We also study the motion of uncharged and charged particles in the RBH background. We determine shadows of such black holes (BHs) using the effective geometry governing motion of photons in the background of NED RBHs. The analysis of circular motion enables to determine innermost stable circular orbits (ISCO) and marginally bound orbits (MBO). We show that the radius of ISCO and MBO for neutral particles decreases as the parameters of the RBH Q and n increase for the fixed value of the BH mass. We demonstrate that for the electrically charged particles properties of the circular orbits strongly depend on their Coulomb interaction with the RBH charge. The dependence of the ISCO radius on the particle specific charge q and the RBH parameters is rather complex, but for the attractive Coulomb interaction there is a general feature (independent of n) giving a limiting value of the intensity of this interaction qQ(max) behind which no stable circular orbits are allowed around the RBH. Comparison of the RBH with Reissner-Nordstrom black holes (RNBH) shows that for the same electric charges of these backgrounds the location of the relevant orbits is at substantially smaller radii for the RBH, demonstrating thus a strong influence of the NED effects. The analysis of ISCO radius of test particles has shown that the charge of RBH Q can mimic the rotation parameter of Kerr BH up to the value a = 0.8M. We have also shown that the RN BH charge can mimic the rotation parameter up to a = 0.5M and the RBH charge up to Q = 0.2M. Applications from observational data to the parameters of the supermassive black holes (SMBHs) at the galactic center Messier 87 (M87) and Milky Way so called Sagittarius A* (Sgr A*) give the mimic value for the RBH charge parameter for the rotation parameter of M87 to be 10(3) Q approximate to 277.56(-1.26)(+3.82), while for Sgr A* there is 10(3) Q approximate to 187.33(-26.4)(+27.6). Moreover, it is shown that the RN BH charge cannot mimic the rotation parameter of the SMBH M87, however, it can mimic the spin parameter of Sgr A* at 10(3)Q(RN)/M = 828.71(-60.94)(+70.58).