Is it permissible for a breastfeeding woman to break her fast in Ramadan, believing it to be permissible, and what is the ruling on her breaking her fast due to her husband inserting his hand into her vagina, and is she required to complete the rest of that day's fast?

A breastfeeding woman is not permitted to break her fast unless she fears for herself or her child. It is not permissible to intentionally invalidate the fast of Ramadan. However, there is no sin upon you and your husband if you are excused due to ignorance. Your fast is invalid according to the majority of scholars who hold that the fast is invalidated by the entry of anything into the vagina, and the safer approach is to make up the day. According to the view that your fast is invalid, you must abstain from eating for the rest of the day and make up that day. As for doubt regarding the emission of semen, nothing is entailed by it, as the default is the validity of the fast, and it is not invalidated by doubt.