In the proof, you have begun by implicitly saying "Assume that the sequence converges, and call the limit $x$". For $1/2 + 1/4 + 1/8 + \cdots$, you obtain $x = 1$. For $2 + 4 + 8 + \cdots$, you obtain $x = -2$.
But that means you have only proved "If $2 + 4 + 8 + \cdots$ converges, then it converges to $-2$". This does not show that the series $2 + 4 + 8 + \cdots$ actually converges. That has to be established separately.
In the case of $1/2 + 1/4 + 1/8 + \cdots$, you can prove by induction that the partial sums are bounded by $1$, and then use the monotone convergence theorem to prove that the series converges. Until you prove the series converges, you don't actually know from your previous calculation that the limit of $1/2 + 1/4 + 1/8 + \cdots$ is $1$. You only know that if $1/2 + 1/4 + 1/8 + \cdots$ converges, then the limit is $1$.
Now, you can't prove the series $2 + 4 + 8 + \cdots$ converges, because it doesn't. But you can prove that if it did converge the limit would be $-2$ - which is still consistent with the series not converging. This is all that your calculation shows.
So the method you are using is fine, but you have to remember that it is relatively meaningless to try to compute the value of a series if the series doesn't converge. When you use the method in the question to compute the value of a series, you have to prove separately that the series converges.