There are no references to Sirius' birthday. He's dead (apparently) and I think JKR doesn't bother about giving dead characters birthdays, no matter how much we writers might need one. I usually make him a Scorpio.

This chapter was great. The idea about Sirius being a shade is quite interesting and original. (Yes, I am aware that was the last chapter, but I got caught up in the whole 'cowboy romance' thing... :D )