Is it obligatory for a woman who performed Umrah while menstruating to repeat it, and is her Hajj valid if she performed Hajj before repeating it?

You erred by prioritizing shyness from people over shyness from Allah. Your circumambulation (Tawaf) while menstruating is not valid according to the majority of scholars. Therefore, you are still in your state of ihram, according to the preponderant opinion. You must refrain from the prohibitions of ihram, then go to Mecca to perform Tawaf, Sa'i, and shorten your hair (taqsir). Alternatively, you may combine Hajj with your Umrah, becoming a Qarinah (one performing Qiran), and then exit your ihram from both after completing your Hajj. If it is difficult for you to return to Mecca, you are considered a Muhsar (one prevented from completing the pilgrimage) and may exit your ihram by sacrificing an animal (hady) or fasting ten days. However, according to the Hanafis, your Umrah is valid, and a camel (badanah) is incumbent upon you.