Problem with solving the recurrence $T(n) = 2T(\sqrt n) + \log n$
Regarding your 1st problem:
You don't need to change $m$ to $\log m$. Let's suppose $f(x)=g(y)$.Then you do not need substitute $y$ anywhere. Because $y$ if substituted would again produce $y$. If you  substitute $y$, you must substitute  it by $g^{-1}f(x)$=$g^{-1}g(y)=y$. You can see you are again getting $y$.

Regarding your second problem:
$S(m)=T(2^m)$

Let $y=m/2$

$$S(y)=T(2^y)$$

$$S(m/2)=T(2^{m/2})$$

You must substitute $m$ by $m/2$. What you are doing is dividing $2^m$ by $2$ which is completely wrong.