How often will the same date be on the same day of the week? Is there any pattern in this (e.g. Once per five years)? 
Is it a sliding window?

All depends on the year it starts first birthday after you been born. If your birthday starts on same day , same date, same month. It will take 5 years,  6 years or 11 years which will bring you to your original day of birthday and the sequence never change. Sequence (11 6 5 6 11 6 5 6 11) all depends on which years sequence(5 6 11 6 5 6 11 6 5)or(6 5 6 11 6 5 6 11 6) you start your sequence, but it never change.You can test it yourself and see the consistency of the sequence birthday, same day of the week.
