Is there an authentic Hadith stating that Allah frees numbers of people from Hell every night of Ramadan, and on the last night, He frees as many as He has freed throughout the entire month?

There is no authentic hadith with the exact wording mentioned, but its meaning has been conveyed in both authentic and weak hadiths. Among the authentic hadiths are:

The hadith: "Indeed, Allah has those whom He frees [from Hell] at every breaking of the fast, and that is every night," narrated by Abu Umamah and Jabir. The hadith: "When the first night of the month of Ramadan comes, the devils are shackled... And Allah has those whom He frees from the Fire, and that is every night," narrated by Abu Hurayrah.

As for the hadith: "When the first night of the month of Ramadan comes, the gates of Paradise are opened... And Allah Almighty has, at every breaking of the fast in the month of Ramadan, every night sixty thousand whom He frees from the Fire..." narrated by Ibn Mas'ud, Al-Albani declared it weak.