On $NP=coNP$ problem in average case
You can construct artificial examples by padding. For example, take an NP-hard language $L$ and pad it by zeroes to $L'=\{x0^{|x|^2}|x\in L\}$. Then the original instances form only a $2^{-n^2}$ fraction, and the answer to the fraction $1-2^{-n^2}$ of the instances is trivially "no" (no certificate is needed at all), yet the language remains NP-hard.

If you look specifically at SAT and the uniform distribution, you need to specify a particular encoding of formulas into bit strings (it will change the effective distribution on formulas). See, for example, the paper by Frieze, Goerdt, and Krivelevich on easy random SAT instances and short proofs of unsatisfiability.