The US extradites people, including US citizens. That is required from the federal government by extradition treaties that are signed voluntarily by the US (in exchange for extradition _to_ the US). It is uncommon because they require escalations through the Department of State and not many crimes are serious enough to justify extradition.
The US does not allow extradition to other countries for crimes, or the enforcement of civil judgements originating in other countries, that stem from conduct that would be constitutionally protected in the US. For example, a US citizen cannot be extradited to face a charge of blasphemy.
No, there is not a precedent for that AFAIK. But the US does extradite, as we don't have "rights guaranteed to us in the United States that no other country has".
...such as? For that matter, you don't think other countries have rights guaranteed to their citizens that the US doesn't have? The legal mismatches don't seem to be likely to make a difference here that could introduce a bias in extraditions.