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We present a spectroscopic investigation with VLT/X-shooter of seven candidate extremely strong damped Lyman-$\alpha$ absorption systems (ESDLAs, $N(\text{HI})\ge 5\times 10^{21}$ cm$^{-2}$) observed along quasar sightlines. We confirm the extremely high column densities, albeit slightly (0.1~dex) lower than the original ESDLA definition for four systems. We measured low-ionisation metal abundances and dust extinction for all systems. For two systems we also found strong associated H$_2$ absorption $\log N(\text{H$_2$)[cm$^{-2}$]}=18.16\pm0.03$ and $19.28\pm0.06$ at $z=3.26$ and $2.25$ towards J2205+1021 and J2359+1354, respectively), while for the remaining five we measured conservative upper limits on the H$_2$ column densities of typically $\log N(\text{H$_2$)[cm$^{-2}$]}<17.3$. The increased H$_2$ detection rate ($10-55$% at 68% confidence level) at high HI column density compared to the overall damped Lyman-$\alpha$ population ($\sim 5-10$%) confirms previous works. We find that these seven ESDLAs have similar observed properties as those previously studied towards quasars and gamma-ray burst afterglows, suggesting they probe inner regions of galaxies. We use the abundance of ionised carbon in excited fine-structure level to calculate the cooling rates through the CII $\lambda$158$\mu$m emission, and compare them with the cooling rates from damped Lyman-$\alpha$ systems in the literature. We find that the cooling rates distribution of ESDLAs also presents the same bimodality as previously observed for the general (mostly lower HI column density) damped Lyman-$\alpha$ population.