Axioms of Minimum Size Resolution Refutations
With the caveat that I am posting this quickly in a sleep-deprived state, I think the answer is "no" to all three questions.

Take the pigeonhole principle formulas PHP^m_n for m pigeons and n holes. The miniminal length of a resolution refutation for m = n+1 is exp(Omega(n)) by Haken. However, Buss and Pitassi proved that for m = exp(\sqrt(n log n)) pigeons (or something in that range) there are refutations of length exp(\sqrt(n log n)) (or something in that range). Clearly, then, the shortest resolution refutation in this case is using lots of redundant clauses (from the point of view of minimal unsatisfiability).