The Qur'an seems to say, over and over, that if you go to Hell, it's forever. From where does the doctrine come that you might be released?

The Quran is the word of Allah , it can not be contradicted. I have personally never read anything about that, just heard it from people.

If the Quran says forever nothing can change that, because Allah is not lying. Allah will remove the bad deeds of those who are permitted into Heaven, doesn't the Quran say that too? So if the good we do outweigh the bad, the bad will be forgiven. Correct me if I'm wrong. The point is that we should believe Allah when he says something. Otherwise we would disbelieve in his words. We can not attribute anything to him if it's not there, people can go to Hell for that. So we should be careful. The Quran can not be contradicted by something else, then the other information is false.

The Noble Quran indicates:

Those who are wretched shall be in the Fire: There will be for them therein (nothing but) the heaving of sighs and sobs: They will dwell therein for all the time that the heavens and the earth endure, except as thy Lord willeth: for thy Lord is the (sure) accomplisher of what He planneth. (11:106-107 - Noble Quran)

"except as thy Lord willeth"

is talking about sinful Muslims, based on ahadith and the Islamic scholarly interpretation.