Mailing a relevant researcher before starting to think about the problem
I would not email people before thinking about a problem. People are busy, so you need to respect that. Instead I would do the thinking and if you have concrete questions, then you can contact others.

If you want to "brainstorm" with someone, you can try to attend a conference that that person also attends. You can sent an email before, and ask if (s)he wants to set half an hour aside for you to discuss this or that problem. Again it would help if you narrowed the problem down, such that it can be discussed in half and hour and the person can prepare a bit if (s)he wants to.