The answers are as follows: these arguments are correct, the indicated function is not needed.

Indeed, your argument might as well prove that for all $H_1$,…,$H_n$, and $H$ such that $c_F(H_1,…,H_n)≤c$, you have $∥P_n⋯P_2P_1−P_0∥≤g_n(c)$.
Quantifying over a collection of sets is perfectly legitimate even if this collection of sets forms a proper class.
There is no need to mention the set $A_n(c)$ at all, it only creates further confusion.