We present a multi-wavelength study of Extremely Red Objects (EROs) employing deep RIzJHK photometry of a 8.5'x8.5' region to identify 68 EROs with R-K>5.3 and K<20.5 (5-sigma). This is combined with an extremely deep 1.4-GHz radio map (sigma=3.5uJy), sensitive enough to detect an active galaxy with L_1.4>10^23 W/Hz at z>1 or a SFR of >25Mo/yr. We detect radio emission from 21 EROs at >12.6uJy and resolve a third of these with our 1.6'' FWHM beam. The SEDs of most of these radio EROs are consistent with dust-reddened starbursts at z~1. At z~1 the radio luminosities of these EROs indicate far-infrared luminosities of L_FIR>10^12 Lo, meaning half are ultraluminous infrared galaxies. We conclude that >16+/-5% of EROs with K<20.5 are luminous infrared galaxies at z>1. We also photometrically classify the EROs to investigate the mix of dusty/active and evolved/passive systems in the radio-undetected EROs. We suggest that at least 30%, and perhaps up to ~60%, of all EROs with R-K>5.3 and K<20.5 are dusty, star-forming systems at z>1. The SFD in this optically faint (R>26) population is rho^* (0.1-100Mo)=0.11+/-0.03 Mo/yr/Mpc^3, comparable to that in H-alpha emitting galaxies at z~1, and greater than that in UV-selected samples at these epochs. This support the claim of a strong increase in obscured star formation at high redshifts. Using the observed counts of the radio-detected EROs we model the break in the K-band number counts of all EROs at K~19-20 and propose that the passive ERO class dominates the total population in a narrow range around K~20, with dusty EROs dominating at fainter magnitudes. [Abridged] Comment: 19 pages, 7 figures, to appear in ApJ Dec 20 2002 v581 n2 revised to comply with proof copy