I don't know if you saw this on the wiki, but if not this may help a little: Life in the Sirdabi Caliphate. This does apply largely to Azadi and therefore Sirdabi cultural practices.
It doesn't have anything about birthdays, though! Birthdays are certainly celebrated throughout the caliphate, though this is somewhere that traditions would definitely vary extremely widely even within Sirdabi culture. The most typical ages for birthday celebrations are at three or four years (when a child has survived many of the greatest perils of early childhood), seven or eight years (typically when a child starts to take on more responsibilities, including the proper observance of Solitary), and sixteen years (when a youth is considered close to an adult). Celebrations are usually a combination of festive good cheer with a more solemn acknowledgement that the child is now moving into a different and steadily more adult phase of life. There is usually some kind of special food involved, the singing of songs, and often small gifts as well.
Coming of age events are even more important, unsurprisingly taking place in the years of early adolescence. This is another area where traditions can differ hugely, between rural and city and nomadic cultures, between rich and poor, between traditionalists and the more modern-minded, and from region to region. Wherever these events are observed however -- and that's nearly everywhere -- it's always the case that these youths are considered to have made an irrevocable transition from childhood to the first stage of adulthood, and their behavior is expected to change accordingly. Usually this means conforming more closely to the stereotypical roles expected of women and men in behavior and dress (though of course this also varies), and a growing concentration on mastering adult skills. Many young people receive a gift deemed suitable for one entering on adulthood, such as a weapon for a male, or a set of implements for cooking or sewing for females. Among the Sirdabi both men and women often receive a jambiya at their coming of age.
This definitely does not answer all your questions but I hope it gives a tiny bit more to work with for the time being!
