concerning your first question, I think that Franz's proof is really in the spirit of Fermat's techniques. Concerning the second question, here is a short, elementary proof, inspired by a theorem of Thue (cf. exercice 1.2 in Franz's book Reciprocity laws). I tried to write it using only notions known to Fermat.
We want to prove that if there exist positive integers $m,n,a$ and $b$ such that $$m^2n=a^2+b^2.$$ $m^2n=a^2+b^2,$$ then $n$ is itself the sum of two integer squares. It is easily seen that it is sufficient to prove this assertion under the hypothesis that $a$ and $b$ (and therefore $a$ and $n$) are coprime and that $n$ is not a square. Let $t$ be the unique positive integer such that $t^2< n<(t+1)^2$. Since there are $(t+1)^2>n$ integers of the form $au+bv$ with $0\leq u,v\leq t$, it follows that $n$ divides the difference $a(u-u')+b(v-v')$ of two of them. Setting $x=u-u'$ and $y=v-v'$ y=v-v'$, we then have the inequalities $|x|,|y|\leq t$. The integer $n$ then divides $a^2x^2-b^2y^2$; since it also divides $a^2y^2+b^2y^2$, it divides their sum, which is equal to $a^2(x^2+y^2)$. Now, the integrs integers $a$ and $n$ being coprime, it follows that $n$ divides $x^2+y^2$. The inequalities $0< x^2+y^2<2n$ finally imply that $n=x^2+y^2$.