$\newcommand{\str}{\{0,1\}}$When trying to tackle such questions, hybrid argument is your friend.
Let $H_0$ denote the real experiment where the output $(w,u,v)$ is generated according to $G'$ -- i.e., for $s\leftarrow\str^n$, $$(x,y):=G(s),(u,v):=G(x),w:=x\oplus y;$$ let $H_2$ denote the random experiment where $(w,u,v)\leftarrow\str^{3n}$.
Now, consider a hybrid experiment $H_1$ where we choose $(x,y)\leftarrow\str^{2n}$ and then set $$(u,v):=G(x),w:=x\oplus y.$$
First, let's prove that $G'$ is a PRG using hybrids.
Using pseudorandomness of $G$, it can be shown that $H_0\approx H_1$, where $\approx$ denotes computational indistinguishability: given a challenge $(a^*,b^*)$ from the PRG experiment of $G$, the reduction simply sets $(x,y):=(a^*,b^*)$. In case $(a^*,b^*)$ is real, then the reduction is simulating $H_0$; otherwise, the reduction is simulating $H_1$.
Similarly, it can be shown that $H_1\approx H_2$: the reduction now sets $(u,v):=(a^*,b^*)$ (here you need your observation that $\oplus$ preserves uniform randomness).
Because indistinguishability is transitive, we get that $H_0\approx H_2$.
Now, when you try something similar with $G''$, you'll see that it doesn't quite go through.
Specifically, the reduction for $H_0\approx H_1$ still works, but it is not clear how to show $H_1\approx H_2$ since it requires a "weird-looking" requirement that, for $y\leftarrow\str^n$, $(G_L(y)\oplus y, G_R(y))$ is pseudorandom, where by $G_L$ and $G_R$, I denote the left and right halves of $G$'s output.
This should already set your alarm bells ringing and you should start thinking about counter-examples to $G''$ being a PRG.
That is, is it possible to design a $G$ such that this property fails?
I'll leave that to you (hint: design $G$ such that it leaks the seed).
