When you get thyroid testing done, insist that they check for the actual, biologically active thyroid hormones, free T3 and free T4 (that's FT3 and FT4, not total T3 and total T4). I say insist because some doctors resist testing anything other than TSH, even though TSH is inadequate as a diagnostic, by which to diagnose and medicate a thyroid patient.
Some will test for FT4, but few want to test for FT3, even though FT3 is the most active thyroid hormone and you really need to be sure of its level. Doctors like to believe that if they know the level of FT4, then they don't kneed to check FT3 because most people convert T4 to T3 adequately. My opinion is why not test and be sure, since FT3 is the most active thyroid hormone, and it largely regulates metabolism and many other body functions, plus it correlates best with hypo symptoms. TSH does not correlate well at all with hypo symptoms.
If your hypothyroidism is well treated then you should get your period monthly like a normal person.
It happened to me with the birth control and all that. Once thyroid medicine works you become regular.
Remember that birth control pill (I assume is that what you are using) is also hormones and that will affect how your thyroid medicine works in your body. You might need more or less thyroid medicine once you stop the pill.
You will need some bloodwork to see where your thyroid and sex hormones levels are and probably readjust the dose.