(1)
$\begin{align}\frac{1+x}{1-x}&=(1+x)\frac{1}{1-x}\\
&=(1+x)\sum_{n=0}^\infty x^n\space(|x|<1)\\
&=\sum_{n=0}^\infty (1+x)x^n\space(|x|<1)\\
\end{align}$
But when $x=-1$, $\sum_{n=0}^\infty (1+x)x^n$ converges
So convergence interval is [-1,1)
(2)
$\begin{align}\frac{1+x}{1-x}&=\frac{1}{1-x}+\frac{x}{1-x}\\
&=\sum_{n=0}^\infty x^n + \sum_{n=0}^\infty x^{n+1} \space(|x|<1)\\
&=1+\sum_{n=1}^\infty x^n + \sum_{n=1}^\infty x^{n} \space(|x|<1)\\
&=1+2\sum_{n=1}^\infty x^n \space(|x|<1)\\
\end{align}$
So convergence interval is (-1, 1)
Series representation of (1) and (2) is different and convergence interval is different. Is this generally correct?
It is correct, if you see them as series of functions. But you should be aware of the fact that your first series is not a power series, whereas the second one is.