Yes, it is a classical result.

Let $q_1,\dots,q_n$ be a sequence of critical points such that $$|q_{n+1}-p|\ge 2\cdot |q_n-p|.$$ By Toponogov's comparison, $$\measuredangle [p\,^{q_i}_{q_j}]\ge \tfrac\pi3.$$ Hence we get a bound on $n$ and hence a bound on $|p-q_n|$ in terms of $|p-q_1|$.

Yes, it is a classical result.

Let $q_1,\dots,q_n$ be a sequence of critical points such that $$|q_{n+1}-p|\ge 2\cdot |q_n-p|.$$ By Toponogov's comparison, $$\measuredangle [p\,^{q_i}_{q_j}]\ge \tfrac\pi3.$$ Hence we get a bound on $n$.