Suppose (counterfactually) that counterfactual Omega asked counterfactual you what factual Omega should write in the factual test (ignoring what factual you actually does, of course). Should the answer (the instruction to Omega to write either "even" or "odd") be the opposite in this counterfactual case than it the case you originally presented?

So far, shokwave clearly gets it. Compare to any other sophisticated question asked in a language you aren't familiar with. Here, you need to be sufficiently comfortable with counterfactuals, for the number of its usages in a problem statement not to act as a pattern for ridiculousness.

it seems that you think that the result on the calculator affects some kind of objective probability that Q is even - a probability that is the same in both factual and counterfactual worlds.

I don't think that.

All probability is subjective. Evidence observed in one world has no influence on counterfactual worlds where the evidence did not appear.

I don't see how "subjective" helps here. It's not clear what sense of "influence" you intend.