Re: plots,

There is such a thing as overfitting, but overplotting cannot really do much harm, especially at diagnostics stage. A standardized normal probability plot cannot hurt next to your QQ-plot. I find it better to assess the middle of the distribution.

Re: residuals,

I run both standardized and studentized residuals at draft stage and usually end up coding the standardized ones. I don't know what other people actually run, because diagnostics are really coded down in the replication material that I find online.

Re: diagnostics,

For a linear model, I usually add variance inflation factors (with the vif command in Stata) and a few homoscedasticity tests (e.g. with the hettest command in Stata), as well as model decomposition with nested regression to check if the $R^2$ makes any sense.