The galaxy-integrated star formation rate (SFR) surface density measurement (ΣSFR) has been proposed as a valuable diagnostic of the mass accumulation in galaxies given it is more tightly related to the physics of star formation and stellar feedback than other indicators. In this work, we assembled a statistical sample of 230 galaxies observed with JWST in the GLASS and CEERS spectroscopic surveys to estimate Balmer line-based dust attenuations and SFRs (i.e., from Hα, Hβ, and Hγ), along with the UV rest-frame effective radii. We studied the evolution of galaxy SFR and ΣSFR in the first 1.5 billion years of our Universe, from a redshift of z ~ 4 to z ~ 10. We found that ΣSFR is mildly increasing with redshift with a linear slope of 0.16 ± 0.06. We explored the dependence of SFR and ΣSFR on stellar mass, showing that a star-forming main sequence and a ΣSFR main sequence are present out to z = 10. This dependence exhibits a similar slope compared to the same relations at lower redshifts, but with a higher normalization. We find that the specific SFR (sSFR) and ΣSFR are correlated with the [O ¯III] λ5007 Å /[O ¯II] λ3727 Å ratio and with indirect estimates of the escape fraction of Lyman continuum photons; hence, they are likely to play an important role in the evolution of ionization conditions at higher redshifts and in the escape of ionizing radiation. We also searched for spectral outflow signatures in the Hα and [O ¯III] emission lines in a subset of galaxies observed at high resolution (R = 2700) by the GLASS survey, finding an outflow incidence of 2/11 (=20%32%9%) = 20 % 9 % 32 % at z < 6, but no evidence at z > 6 (0/6, < 26%). Finally, we find a positive correlation between AV and ΣSFR, and a flat trend as a function of sSFR, indicating that there is no evidence of a drop in AV in extremely star-forming galaxies between z ~ 4 and ~10. This result might be at odds with a dust-clearing outflow scenario, which may instead take place at redshifts of z ≥ 10, as suggested by some theoretical models.