Motivated by the recent observational and theoretical evidence that long Gamma-Ray Bursts (GRBs) are likely associated with low metallicity, rapidly rotating massive stars, we examine the cosmological star formation rate (SFR) below a critical metallicity Z_crit Z_sun/10 - Z_sun/5, to estimate the event rate of high-redshift long GRB progenitors. To this purpose, we exploit a galaxy formation scenario already successfully tested on a wealth of observational data on (proto)spheroids, Lyman break galaxies, Lyman alpha emitters, submm galaxies, quasars, and local early-type galaxies. We find that the predicted rate of long GRBs amounts to about 300 events/yr/sr, of which about 30 per cent occur at z>~6. Correspondingly, the GRB number counts well agree with the bright SWIFT data, without the need for an intrinsic luminosity evolution. Moreover, the above framework enables us to predict properties of the GRB host galaxies. Most GRBs are associated with low mass galaxy halos M_H<~10^11 M_sun, and effectively trace the formation of small galaxies in such halos. The hosts are young, with age smaller than 5*10^7 yr, gas rich, but poorly extincted (A_V<~0.1) because of their chemical immaturity; this also implies high specific SFR and quite extreme alpha-enhancement. Only the minority of hosts residing in large halos with M_H>~10^12 M_sun have larger extinction (A_V~0.7-1), SFRs exceeding 100 M_sun/yr and can be detected at submm wavelengths. Most of the hosts have UV magnitudes in the range -20 <~M_1350<~ -16, and Lyman alpha luminosity in the range 2*10^40 <~L_Lya<~2*10^42 erg/s. GRB hosts are thus tracing the faint end of the luminosity function of Lyman break galaxies and Lyman alpha emitters.
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