What is the name of the cycle whereby the lunar azimuth and altitude recur on a given day?
You are looking for the Metonic cycle.  This is almost exactly 19 years, and represents 235 synodic months.

Now you have edited the detailed question so that it is out of sync with the original Title.  The other period you need is the Draconic month, which is the period between the moon crossing the ecliptic (in the same direction).