Does divorce occur in the case of swearing by "I divorce my wife" without the intention of divorce, and then sitting on the sworn-upon item out of forgetfulness? And is an oath expiation required, or is there nothing upon the one who did that?

If you swore by divorce that you would not sit on a chair, and then you sat on it forgetfully, then according to the Hanbalis and the well-known opinion among the Malikis, divorce takes effect. This is in contrast to the Shafi'is and some Malikis. You are permitted to follow the madhhab (definition) of the Shafi'is and some Malikis, in which case neither divorce nor expiation is binding upon you. As for the opinion that an expiation for an oath (kaffarat yamin (definition)) is required in this case, it is the view of Sheikh al-Islam Ibn Taymiyyah, if the husband did not intend divorce but rather intended an oath or a threat.