On $NP=coNP$ problem in average case $1)$Can a $NP$-hard problem can have short certificates that can be verified in deterministic polynomial time for ALL but $\frac{1}{n}$ of the NO instances$?

$2)$ Can one concoct versions of NP complete problems $\Pi_1$ and $\Pi_2$ where probability $p=P(\Pi_2=NO|\Pi_1=YES)=P(\Pi_1=NO|\Pi_2=YES)$ is between $\frac{1}{2}+\epsilon$ and $1-\epsilon$? Then almost all the time short certificates are possible as in $1)$ if $p$ is closer to $1-\epsilon$.
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.