We report results from a programme aimed at investigating the temperature of neutral gas in high-redshift damped Lyman-$alpha$ absorbers (DLAs). This involved (1) HI 21cm absorption studies of a large DLA sample, (2) VLBI studies to measure the low-frequency quasar core fractions, and (3) optical/ultraviolet spectroscopy to determine DLA metallicities and velocity widths. Including literature data, our sample consists of 37 DLAs with estimates of the spin temperature $T_s$ and the covering factor. We find a strong $4sigma$) difference between the $T_s$ distributions in high-z (z>2.4) and low-z (z<2.4) DLA samples. The high-z sample contains more systems with high $T_s$ values, $gtrsim 1000$ K. The $T_s$ distributions in DLAs and the Galaxy are also clearly (~$6sigma$) different, with more high-$T_s$ sightlines in DLAs than in the Milky Way. The high $T_s$ values in the high-z DLAs of our sample arise due to low fractions of the cold neutral medium. For 29 DLAs with metallicity [Z/H] estimates, we confirm the presence of an anti-correlation between $T_s$ and [Z/H], at $3.5sigma$ significance via a non-parametric Kendall-tau test. This result was obtained with the assumption that the DLA covering factor is equal to the core fraction. Monte Carlo simulations show that the significance of the result is only marginally decreased if the covering factor and the core fraction are uncorrelated, or if there is a random error in the inferred covering factor. We also find evidence for redshift evolution in DLA $T_s$ values even for the z>1 sub-sample. Since z>1 DLAs have angular diameter distances comparable to or larger than those of the background quasars, they have similar efficiency in covering the quasars. Low covering factors in high-z DLAs thus cannot account for the observed redshift evolution in spin temperatures. (Abstract abridged.)
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