Your thyroid levels are pretty normal. I don't think an increase in dose would help much, but you could consider an extra 1/2 tab per week after discussing with prescribing doctor -- however, research suggests that such "fine tuning" does not help symptoms.
Change in estrogen doses in BCP may alter thyroid requirements, otherwise not likely connected.
Your numbers all look normal, if that is free T4. You should request the lab ranges with those results so you can see for yourself.
Don't know about the hormones in the bc pills.
You should get the range from your blood test report. Your numbers may be normal in general, but each person is different. I need very low TSh (somewhere around 0.3) and high normal of T3 and T4 to feel better. good luck..
Curious about the Yaz. I have been on it a few months and just started SYnthroid last week. It seems that the PMS this month is much worse than it has been as the Yaz had all but eliminated most of it until now. Wondering if its the Synthroid interacting.