While working on the first question, i managed to get another solution, different from Mark's. Consider the formal product: $$P_0=\prod_{p_i \, prime} p_{i}^{p_i}.$$ Note that it is the fixed point of the formal derivation. Then the formal derivative of $2P_0$ is $f(2P_0)=3P_0$. In general it is true that $f^{(k)}(2P_0)=c_k f^{(k-1)}(2P_0)$ for a $c_k>1$. Indeed as every exponent is at least as big as its base. Thus lets formally derive $2P_0$, k-times. Then there are only finitely many primes which had their exponents changed during the process. Removing every other prime from $2P_0$ we obtain a finite product which has its first $k$ derivatives strictly increasing.

While working on the first question, i managed to get another solution to the second, different from Mark's. Consider the formal product: $$P_0=\prod_{p_i \, prime} p_{i}^{p_i}.$$ Note that it is the fixed point of the formal derivation. Then the formal derivative of $2P_0$ is $f(2P_0)=3P_0$. In general it is true that $f^{(k)}(2P_0)=c_k f^{(k-1)}(2P_0)$ for a $c_k>1$. Indeed as every exponent is at least as big as its base. Thus lets formally derive $2P_0$, k-times. Then there are only finitely many primes which had their exponents changed during the process. Removing every other prime from $2P_0$ we obtain a finite product which has its first $k$ derivatives strictly increasing.