I think it's a reasonable question. The quick answer is that there's nothing wrong with emailing someone you think might want to work with you. The first email should be brief, polite, and give a very short explanation of who you are. ("I graduated with a master's from X university in 201X and continue to have an interest in subject X.") Then it should give an again, very brief description of the problem and what you hope to achieve.

To your other question, I myself left academia eight years ago because of a two-body problem, and the biggest surprise for me was that MathSciNet was no longer available to me. Not that an individual subscription would cost an exorbitant amount --- I was directly told by the AMS that if I am not affiliated to an institution, there is no way to subscribe, period. The best recommendation I can give you here is to look for SciHub, if I am allowed to say the name directly.