What is the ruling on the fasting and expiation for a woman who broke her fast for several consecutive Ramadans due to studying abroad and does not know their exact number? And is it permissible for a person who is not a religious scholar to invite people to Islam by publishing Quranic verses in English on social media?

Intentionally breaking the fast in Ramadan (definition) is a major sin. Whoever does this must repent and make up the days they intentionally missed. If they do not know the exact number, they should investigate until they are reasonably certain that they have fulfilled their obligation. As for da'wah (calling to Allah), it is a great act that requires sufficient religious knowledge. There is no harm in sharing video clips of translated verses, hadiths, stories of the prophets, and the like, provided that they come from reliable sources and that the information's authenticity is verified by presenting it to scholars.