Does the questioner's asking the Sheikh about menstruation issues, after he had asked her not to ask him and to suffice with asking the female muftiya, constitute breaking a promise and entering into one of the characteristics of hypocrites?

What you did by breaking the promise — if you had committed not to ask him again — there is no sin in it, as long as you intended to fulfill it at the time of your speaking, and then you saw a benefit in breaking it. This is not among the characteristics of hypocrites, because breaking a promise is blameworthy only if the intention to break it accompanies the promise. However, if one intended to fulfill it and then an impediment arose or a different opinion occurred to him, then the characteristic of hypocrisy has not manifested from him. It has been reported in the hadith: "If a man promises his brother, and his intention is to fulfill it, but he does not fulfill it, there is no sin upon him."