Is the Hajj valid after redoing the Sa'i and Tawaf, with intercourse occurring during the period between the first Sa'i, which was believed to be valid, and the second Sa'i, and sufficing with feeding six needy persons as expiation for that?

It is clear that your Hajj is valid, and you have fulfilled your obligation through the circumambulation (Tawaf) and the striving (Sa'i). As for sexual intercourse, there is no penalty upon you for it due to ignorance, according to many scholars, and this is the relied-upon opinion. However, according to the Hanbalis, a fidyah (expiation) is required (feeding six needy persons, or fasting three days, or sacrificing a sheep), along with returning to ihram from outside the Haram area (Al-Hill). Yet, the majority of scholars do not obligate going out to Al-Hill. And your feeding of the six needy persons is sufficient according to the Hanbalis, and there is no harm upon you in adopting the opinion of those who do not obligate going out to Al-Hill.