Context: On the nature, redshift, stellar populations and dust properties of optically faint or non-detected extremely red objects. Aims: Determining the nature, redshift, stellar populations and dust properties of optically faint or non-detected, extremely red objects (ERO) found from our survey of the lensing clusters A1835 and AC114 (Richard et al. 2006, A&A, 456, 861). Comparison with properties of related galaxies, such as IRAC selected EROs and a z 6.5 post-starburst galaxy candidate from the Hubble Ultra Deep Field. Methods: Using an updated version of Hyperz (Bolzonella et al. 2000, A&A, 363, 476) and a large number of spectral templates we perform broad-band SED fitting. The photometric observations, taken from Hempel et al. (2007, A&A, submitted), include deep optical, ACS/HST, ISAAC/VLT, IRAC/Spitzer data, and for some objects 24 ?m MIPS/Spitzer and sub-mm data as well. Results: For most of the lensed EROs we find photometric redshifts showing a strong degeneracy between ?low-z? (z 1-3) and high-z (z 6-7). Although formally best fits are often found at high-z, their resulting bright absolute magnitudes, the number density of these objects, and in some cases Spitzer photometry or longer wavelength observations, suggest strongly that all of these objects are at ?low-z?. The majority of these objects are best fitted with relatively young (?0.5-0.7 Gyr) and dusty starbursts. Three of our objects show indications for strong extinction, with AV 2.4-4. The typical stellar masses of our objects are M_star (0.5-5)×1010 M? after correction for lensing; for the most extreme ERO in our sample, the sub-mm galaxy SMMJ14009+0252 most likely at z_fit 3, we estimate M_star ? 6.×1011 M?. For dusty objects star formation rates (SFR) have been estimated from the bolometric luminosity determined after fitting of semi-empirical starburst, ERO, and ULIRG templates.Typically we find SFR (1-18) M? yr-1. Again, SMMJ14009+0252 stands out as a LIRG with SFR 1000 M? yr-1. Finally, we predict the mid-IR to sub-mm SED of the dusty objects for comparison with future observations with APEX, Herschel, and ALMA. Concerning the comparison objects, we argue that the massive post-starburst z 6.5 galaxy candidate HUDF-J2 showing observed properties very similar to our EROs, is more likely a dusty starburst at z 2.3-2.6. This interpretation also naturally explains the observed 24 ?m emission from this object and we predict its IR to sub-mm SED. Both empirically and from our SED fits we find that the IRAC selectec EROs from Yan et al. (2004, ApJ, 616, 63) show very similar properties to our lensed EROs. Reasonable fits are found for most of them with relatively young and dusty stellar populations. Based on observations collected at the Very Large Telescope (Antu/UT1), European Southern Observatory, Paranal, Chile (ESO Programs 69.A-0508, 70.A-0355, 73.A-0471), the NASA/ESA Hubble Space Telescope obtained at the Space Telescope Science Institute which is operated by AURA under NASA contract NAS5-26555, the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under NASA contract 1407, and the Chandra satellite.