is y[n] = x[n] + n time invariant?
No; the system given by

$$ y[n] = x[n] + n $$ is time-varying, due to the added term $n$.

Your mistake is in the line :
$$\ y_2[n] = x_2[n] + n = x[n-k]+(n-k) $$ which should be instead

$$\ y_2[n] = x_2[n] + n = x[n-k]+n$$ and therefore implies that

$$y_2[n] \neq y[n-k] $$.