>>> Is the spearman test run in stata corrected for ties?
In -spearman- the correlation is just the Pearson correlation
calculated on the ranks. Thus Stata does nothing special about
ties. I think Stata here shares in a widespread prejudice: if
ties are an issue for you, then either your sample is too small
or the data are too problematic for Spearman to be appropriate.
Getting the P-value exactly right if the main idea is dubious
is putting the emphasis in the wrong place. No doubt other
software can fill the gap.
Roger Newson's -somersd- may be better for your purpose.
>>> What does it mean ASE when computing the value of the Goodmans and
Kruskals gamma? Is it the significance level? If not, how to define the
significance of the correlation?
(Almost surely erroneous?) Asymptotic standard error.