Is it necessary to make up for the day on which vomit returned to the stomach before reaching the mouth, and the day on which a very small piece of skin was swallowed unintentionally, according to the Hanafi school of thought?

Your fast is valid as long as the vomit did not reach a place from which it could be expelled. Swallowing lip skin or anything else unintentionally does not invalidate the fast according to the majority of scholars. It is stated in Kanz al-Daqaiq by al-Nasafi al-Hanafi that what does not invalidate the fast includes one who eats or drinks forgetfully, or if dust or a fly enters their throat, or if they eat what is between their teeth, or if they vomit and it returns. Therefore, you do not need to make up the fast.