You don't need to use sureg. The benefit of SURE is smaller standard errors, but you could just estimate separate regressions. This is what I would advise. Are the independent variables the same in each regression? If so, then there is no efficiency gain in SURE at all, and separate regressions is what you should use.
Your syntax for sureg seems incorrect, ...
Just regress Y on X:
and you will likely find some negative significant $b_1$ coefficient even though both series are just unrelated random walks.
You can also see that as one series increases other one decrease so you would expect they are correlated in negative way in this case.
Yes, the null hypothesis of ADF test is that the series contains unit root (e.g. see Verbeek, A guide to modern econometrics pp 273).
So the results you present above indicate that you cannot reject the null of an unit root and consequently you should treat your series as non-stationary.
How can I use the Dickey-Fuller test statistic in this case to ...