Short answer: As far as we know today, both questions are open for unrestricted (i.e., arbitrary-depth) circuits. We do not know any explicit monotone function family in P/poly for which every monotone circuit that achieves even constant-bounded error under the uniform distribution (or at its critical p) must have superpolynomial size; conversely, we also do not have a general positive theorem showing that such monotone approximators always exist.

Some context and related results:

• Exact lower bounds for monotone circuits. Razborov’s classic lower bounds show exponential monotone circuit complexity for natural monotone functions such as CLIQUE and PERFECT MATCHING. However, these address exact computation; they do not by themselves yield average-case (correlation) lower bounds against unrestricted-depth monotone circuits.

• Your matching example. As you noted, perfect matching is a cautionary tale for average case at the critical probability: near its threshold, “no isolated vertices” (a very simple, low-depth monotone property) already gives an excellent approximation. So matching does not refute either question.

• Constant-depth (AC^0) analogues. Here we do have strong average-case lower bounds. For example, for k-Clique and other subgraph/isomorphism-type monotone properties, there are superpolynomial average-case lower bounds for AC^0 circuits (monotone or not) with respect to appropriate Erdős–Rényi distributions, e.g., Rossman’s work on subgraph isomorphism/clique in AC^0. These show that the AC^0 versions of your questions can fail. On the positive side, for fixed depth d, one can often ε-approximate AC^0 circuits by simpler monotone circuits (e.g., via switching-lemma-based arguments), but the size blowups are quasi-polynomial rather than polynomial in general. None of this settles the unrestricted-depth case.

• Lack of techniques for unrestricted-depth average-case bounds. We currently do not have methods to prove superpolynomial lower bounds for correlation with arbitrary-depth monotone circuits for explicit monotone functions in P/poly. Thus, we can’t rule out a positive answer either. Producing any explicit monotone f ∈ P/poly that is average-case hard (say, under μ1/2 or at p = pc(f)) for all polynomial-size monotone circuits would be a major breakthrough.

Summary:

• Question 1 (uniform distribution): Open.
• Question 2 (at p = pc(f)): Open.
• For constant depth, there are strong negative results (average-case lower bounds), but they do not extend to arbitrary depth.
• Matching shows that in specific cases, good monotone approximators exist at the critical probability, but we do not know a general theorem.

References to look at for related context: Razborov’s monotone lower bounds for CLIQUE/MATCHING; Alon–Boppana on monotone circuit complexity; and Rossman’s average-case lower bounds for AC^0 on subgraph isomorphism/clique.