When to give up and ask the research community?
This is a type of question that you could ask on MathOverflow.  

In my experience, one of the best uses of MO is if you have a "research level" question that (i) you want answered, and you're not particular about how and (ii) you think "Probably[/maybe/I wonder if] someone with the right expertise could solve this almost immediately."  

If both of the above apply, I would encourage you to ask on MO.  On the other hand, if you want to keep maximal control over your involvement with the question and its solution, less wide disbursement is probably best.  (Most of my own MO questions have been things that come up in my research but are not exactly the questions I am trying to solve in order to write my next paper.) If you really see yourself getting a paper out of these questions, I would vote for spending more than two weeks thinking them over.