Yes, $N$ is the subgroup generated by all squares. $N$ is normal in $F$, because if $w^2$ is an element of the generating set for $N$, and $x\in F$, then $xw^2x^{-1} = (xwx^{-1})^2$ is also an element of $N$. So for every $x\in F$ we have $xNx^{-1}\subseteq N$, hence $N$ is normal.
When $p$ is a prime, an abelian $p$-group is simply an abelian group all of whose elements have order a power of $p$ (the group could be finite or infinite). An elementary abelian $p$-group is an abelian $p$-group in which every element satisfies $a^p = 1$ (and so, every element except for the identity is of order exactly $p$; this is with multiplicative notation, you would have $pa=0$ if you are using additive notation for your group). So the assertion is that $F/N$ is abelian, and the square of every element is the identity.
The fact that the square of every element of $F/N$ is the identity follows because every square is in $N$: if $fN\in F/N$, then $(fN)^2 = f^2N = 1N$. And the fact that $F/N$ is abelian now follows from the well-trod fact that a group in which the square of every element is the identity must be abelian (since $1 = (ab)^2 = a^2b^2$, so $abab=aabb$, hence $ba=ab$ by cancellation).
An elementary abelian $p$-group is always a vector space over $\mathbb{F}_p$, the field with $p$-elements: given $\alpha\in\mathbb{F}_p$, let $a\in\mathbb{Z}$ by any integer mapping to $\alpha$. Then, assuming your group is written additively, define $\alpha\cdot g$ as $\alpha\cdot g= ag$. Since $pg=0$, this is well defined and makes the abelian group into a vector space.
So here, you have that $F/N$ is an abelian $2$-group, and therefore is a vector space over the field of $2$-elements.
In fact, the images of the free generating set $X$ in $F/N$ form a basis for this vector space: since $X$ spans $F$, its images span $F/N$. And if you have a nontrivial linear combination between them, then it must be of the form
$$\overline{x_1}+\cdots + \overline{x_n} = \mathbf{0}$$
where $\overline{g}$ is the image of $g\in F$ in the quotient, and $x_1,\ldots,x_n$ are pairwise distinct elements of $X$. But this means that $x_1\cdots x_n\in N$, that is, that it is the square of an element of $F$, and this is easily shown to be impossible.
So $F/N$ is a vector space over $\mathbb{F}_2$, and has a basis of cardinality $|X|$. How many elements does a vector space of dimension $\kappa$ over $\mathbb{F}_2$ have? If $\kappa$ is finite, then it has $2^{\kappa}$ elements. If $\kappa$ is infinite, then it has $\kappa$ elements. So if $X$ is infinite, then $|X|=|F/N|$.
Now suppose that $F_X$ and $F_Y$ are isomorphic. Then the isomorphism maps $N=\langle w^2\mid w\in F_X\rangle$ to $M=\langle z^2\mid z\in F_Y\rangle$, so we get that $F_X/N\cong F_Y/M$. If one of them is finite, then they both are; if one of them is infinite, then they both are. If both are finite, then the cardinality of $F_X/N$ is $2^{|X|}$, and the cardinality of $F_Y/M$ is $2^{|Y|}$, and since they are isomorphic groups, then $|X|=|Y|$. If they are both infinite, then $F_X/N$ has cardinality $|X|$, and $F_Y/M$ has cardinality $|Y|$, and since they are isomorphic their cardinalities are the same, so $|X|=|Y|$ as well. This proves the result.
There is a simpler way: the result holds for abelian free groups (tensor up to $\mathbb{Q}$ to reduce to the vector space $K$), and then show that $F_X^{\mathrm{ab}}$ is the free abelian group on $X$.
To answer final question: "word length" depends on the free basis. There is always a choice of basis for $F_Y$ that makes $\varphi$ preserve word length (simply take the basis $\varphi(X)$), but in general it need not. Take $X=\{x_1,x_2\}$, $Y=\{y_1,y_2\}$, and first map $F_X$ to $F_Y$ the obvious way ($x_1\mapsto y_1$, $x_2\mapsto y_2$), and then compose with a suitable inner automorphism of $F_Y$. For example, composing with conjugation by $y_1$ maps $x_1\mapsto y_1$ and $x_2\mapsto y_1y_2y_1^{-1}$. Composing with conjugation by $y_1y_2y_1^{-1}$ makes it even worse:
$$\begin{align*}
x_1 &\mapsto (y_1y_2y_1^{-1})y_1(y_1y_2^{-1}y_1^{-1}) = y_1y_2y_1y_2^{-1}y_1^{-1},\\
x_2 &\mapsto (y_1y_2y_1^{-1})y_2(y_1y_2^{-1}y_1^{-1}) = y_1y_2y_1^{-1}y_2y_1y_2^{-1}y_1^{-1},
\end{align*}$$
mapping the two generators to words of length 5 and 7, respectively; of course, you can make it pretty much as bad as you want using this idea.
