How to decide between sending paper to a specialized journal or to a journal with broader audience?
I tend to think that in most cases, the specialized/broad dichotomy is not very relevant. The most important point is to send it to an interested editor; if the most relevant editor for a paper happens to be at a general journal, you will often be better of sending your paper there.

I would consider two exceptions to this principle. First, top specialized journals are usually less reputed than top generalist journals, so if you get a truly impressive result, you may want to get the best of it by sending it to a top generalized journal. Second, some generalist journals will turn back papers that seems much more specialized than the average math paper (e.g. when the basic objects you study are unheard of by most mathematicians).