Let’s try to work in $\Bbb Z$, at least at first: it’s the most familiar additive group.
Clearly $W(A,B)\supseteq\{\langle a,a,b,b\rangle:\langle a,b\rangle\in A\times B\}$, so $|W(A,B)|\ge N^2$. Thus, to get $|W(A,B)|$ comparable to $N^2$, we need to try to arrange matters so that a+b=a'+b' iff \langle a,b\rangle=\langle a',b'\rangle. Clearly that’s as small as we can get.
Suppose, on the other hand, that for each $\langle a,b\rangle\in A\times B$ and each a'\in A, a+b-a'\in B; then each $\langle a,b\rangle\in A\times B$ would generate $N$ members of $W(A,B)$, and $|W(A,B)|$ would be comparable to $N^3$. It’s not too hard to see that this is as large as we can get.
For the first problem, then, we’d like to choose $A,B$, and $C$ so that
for each $\langle a,b\rangle\in A\times B$ and a'\in A, a+b-a'\in B iff a'=a;
for each $\langle a,c\rangle\in A\times C$ and a'\in A, a+c-a'\in C iff a'=a; and
for each $\langle b,c\rangle\in B\times C$ and b'\in B, b+c-b'\in C.
Suppose that $A=\{1,\dots,N\}$. Then for any a,a'\in A we have -N. Suppose now that we can choose $B$ so that if $b$ and b' are distinct members of $B$, then |b-b'|\ge N. Now suppose that $\langle a,b\rangle\in A\times B$ and a'\in A. If a+b-a'=b'\in B, then a-a'=b'-b; but |a-a'|, and |b'-b|\ge N unless b=b', in which case a=a'. Thus, such a $B$ would satisfy (1). Is there such a $B$? Sure: just let $B=\{kN:k=1,\dots,N\}$.
If we let $C=B$, (2) is automatically satisfied. (3) isn’t satisfied, but it is true that if $\langle b,c\rangle\in B\times C$, b'\in B, and b+c>b', then b+c-b'\in C. How many triples \langle b,c,b'\rangle\in B\times C\times B are there such that b+c>b'?
It’s actually easier to count the triples with b+c\le b'. I leave it to you to verify that there are
$\begin{align*} \sum_{i=1}^{N-1}i(N-i)&=N\sum_{i=1}^{N-1}i-\sum_{i=1}^{N-1}i^2\\ &=\frac12N^2(N-1)-\frac16N(N-1)(2N-1)\\ &=\frac{N(N-1)}6\big(3N-(2N-1)\big)\\ &=\frac16(N^3-N) \end{align*}$
of these, and therefore $\frac56N^3+\frac16N$ triples with b+c>b'. This should qualify as comparable to $N^3$.
Edit: For the second problem, suppose that $N=2n$ and try
$\begin{align*} A&=\{1,\dots,n\}\cup\{kn^2:k=1,\dots,n\},\\ B&=\{1,\dots,n\}\cup\{kn^4:k=1,\dots,n\},\text{ and}\\ C&=\{kn^2:k=1,\dots,n\}\cup\{kn^8:k=1,\dots,n\}\;. \end{align*}$
Try to show that the overlapping parts of $A$ and $B$ and of $A$ and $C$ ensure that $|W(A,B)|$ and $|W(A,C)|$ are at least some fixed multiple of $N^3$, while $B$ and $C$ are related more like my $A$ and $B$ in the first problem.