I have a date range July–September 2015. It doesn't look right. Should it be July–September, 2015? Have you any suggestions, or is it right to begin with?
This question takes us into the area of discretionary punctuation. Different style guides have different preferences on these points, and it seems to me that there is no compelling reason to prefer one guide's approach over another's.
Having said that, let me outline the recommendations that one particular style guide—The Chicago Manual of Style, sixteenth edition (2010)—makes. First, regarding the use of en dashes in regular text, Chicago says this:
6.78 En dash as "to." The principal use of the en dash is to connect numbers and, less often, words. With continuing numbers—such as dates, times, and page numbers—it signifies up to and including (or through). For the sake of parallel construction, the word to, never the en dash, should be used if the word from precedes the first element in such a pair; similarly, and, never the en dash, should be used if between precedes the first element.
[Relevant examples:] I have blocked out December 2009–March 2010 to complete my manuscript.
The London–Paris train leaves at two o'clock.
This guideline asserts, in the first place, that if your construction reads as "from July–September 2015," you should rewrite it as "from July to [or through] September 2015"; and that if it reads "between July–September 2015," you should rewrite it as "between July and September 2015." On the other hand, it seems to endorse the construction "the period July–September 2015," if you want to use an en dash (rather than to or through) there.
Another point that we may infer from the first example is that Chicago sees no reason to insert a comma between a month and a year in terms such as "December 2009" and "March 2010." I am not aware of any logical basis for rendering "December 2009" and "March 2010" without commas but requiring insertion of a comma after the month range in "May–July 2016."
Elsewhere, Chicago has this to say about commas with dates:
6.45 Commas with dates. In the month-day-year style of dates, commas must be used to set off the year. In the day-month-year system—useful in material that requires many full dates (and standard in British English)— no commas are needed. Where month and year only are given, or a specific day (such as a holiday) with a year, neither system uses a comma. ...
[Relevant example:] In March 2008 she turned seventy-five.
It seems clear from this guideline and the one cited earlier that Chicago strongly favors the form "December 2009–March 2010" (with no comma) in any instance where you are using en dash properly in a months-and-year(s) range. But again, Chicago would not approve of prefacing such a range with "from" or "between."