The galaxy integrated star-formation rate (SFR) surface density ($\Sigma_{\rm SFR}$) has been proposed as a valuable diagnostic of the mass accumulation in galaxies as being more tightly related to the physics of star-formation (SF) and stellar feedback than other SF indicators. In this paper, we assemble 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, and UV rest-frame effective radii. We study the evolution of galaxy SFR and $\Sigma_{\rm SFR}$ in the first 1.5 Billion years of our Universe, finding that $\Sigma_{\rm SFR}$ is mildly increasing with redshift with a linear slope of $0.16 \pm 0.06$. We also explore the dependence of SFR and $\Sigma_{\rm SFR}$ on stellar mass, showing that a SF 'Main-Sequence' and a $\Sigma_{\rm SFR}$ `Main-Sequence' are in place out to z=10, with a similar slope compared to the same relations at lower redshifts. We find that the specific SFR (sSFR) and $\Sigma_{\rm SFR}$ are correlated with the [OIII]5007/[OII]3727 ratio and with indirect estimates of the escape fraction of Lyman continuum photons, hence they likely play an important role in the evolution of ionization conditions and in the escape of ionizing radiation. We also search for spectral outflow signatures in a subset of galaxies observed at high resolution, finding an outflow incidence of $2/11$ ($=20\%^{32\%}_{9\%}$) at $z6$ ($<26\%$). Finally, we find a positive correlation between A$_V$ and $\Sigma_{\rm SFR}$, and a flat trend as a function of sSFR, indicating that there is no evidence of a drop of A$_V$ in extremely star-forming galaxies between z=4 and 10. This might be at odds with a dust-clearing outflow scenario, which might instead take place at redshifts $z\geq 10$, as suggested by some theoretical models.

Comment: Accepted for publication in A&A ; 19 pages, 9 figures, 1 table. Compared to the previous version, we have corrected a typo in Fig. 5 (the x and y axis labels were inverted)