Advice for contacting journals for the first time What can be recommended for contacting journals with the purpose of checking whether they would be interested in an publishing an article if one is not affiliated to a research institution and still is unknown to the professional math community?  
Specifically I would like to know whether it is acceptable to initially only state the problem whose solution is claimed and give some indication that there is strong evidence for the correctness of that solution, e.g. by providing experimental results, proving acquaintance with the problem and attempts to solve it.  
Are such initial contacts tolerated by journals or do they insist on the availability of a mature document when they are initially contacted? 
 A: Sending an editor an outline like you describe won't have much of an effect. All an editor will do, if they respond at all, is say "why don't you submit the paper, and then we will decide whether to send it for peer review and to accept it". They couldn't possibly be any more committal based on just an email.
Writing mathematical papers is hard! Graduate students and early postdocs struggle a lot with that, and typically rely on considerable input from their advisors or other more senior colleagues to produce a paper that is submittable.
What I would recommend is that you first find a professional mathematician who is interested in your problem, and contact them with the questions that you would ask an editor. If you cannot find a mathematician who is interested in your problem, then that does not bode well for your journal submission (since if the editor cannot find anybody interested in it, they will reject the paper).
A: If the point is to get published in a journal, you need to do it their way.  Their job is to take mature papers, have them vetted to make sure they are mature enough to commit to the literature, and then publish them with the stamp of their reputation.  If you are unsure whether your paper merits their approval, you probably should not ask.  (This is different from trying to determine publishability.  If you contact an editor or two, they might give you their opinion if it does not take much time.  I think Alex B. has a correct and charitable reading of the situation here.)
If the goal instead is to gauge interest in the problem, there are better ways.  Probably the best is to look up conference proceedings of the area and see how much play the problem is given.  Use these to get names and addresses of experts who can probably tell you how and when to approach journals, if you still want to go that route.  There is less barrier to self publishing, and you can use your MathOverflow user page to talk up about yourself and your work.  The user page can serve as a big business card should you wish to contact these experts.
Gerhard "Don't Need No Stinkin' Journals" Paseman, 2020.06.14.
