How does publication month and number of citations to a paper correlate?
It doesn't appear to be a major factor. On the subject of citation behavior, I recommend reading this very rich article:

Lutz Bornmann and Hans-Dieter Daniel What do citation counts measure? A review of studies on citing behavior Journal of Documentation Vol. 64 No. 1, 2008 pp. 45-80

Generally, it confirms your point 1. Previous citation count is a good predictor of future citation. 

It seems to contradict your point 2. There are more citation to recent article mainly because there are much more recent articles, not because they are preferably cited over older articles. 

I can only address your question and point 3 by default: the month of publication is not listed among the potential factors influencing citation in this study nor in several other quantitative studies (1) (2) (3), although it was not formally excluded.


  Note that an effective data analysis would likely need to exclude very
  highly cited papers that might introduce a bias.


Yes, most likely the effect of the publication month, if there is one, will be masked by a variety of other stronger factors.