Since it is known that $f(N)\sim \sqrt{N}$ (see [30]) the latter question is equivalent to asking whether, if $\lvert A\rvert=\lvert B\rvert$,\[\lvert A\rvert \leq \left(\frac{1}{\sqrt{2}}-c+o(1)\right)\sqrt{N}\]for some constant $c>0$. In the comments Tao has given a proof of this upper bound without the $-c$.

In the comments Barreto has given a negative answer to the second question: for infinitely many $N$ there exist Sidon sets $A,B\subset \{1,\ldots,N\}$ with $\lvert A\rvert=\lvert B\rvert$ and $(A-A)\cap (B-B)=\{0\}$ and\[\binom{\lvert A\rvert}{2}+\binom{\lvert B\rvert}{2}\geq (1-o(1))\binom{f(N)}{2}.\]