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.