  We show that if $X$ has a zero-set diagonal and $X^2$ has countable weak
extent, then $X$ is submetrizable. This generalizes earlier results from Martin
and Buzyakova. Furthermore we show that if $X$ has a regular
$G_\delta$-diagonal and $X^2$ has countable weak extent, then $X$ condenses
onto a second countable Hausdorff space. We also prove several cardinality
bounds involving various types of diagonal degree.
