Is what I did with my married cousin, exchanging nude pictures and asking her to take off her house abaya and touching her hair, considered fornication (zina)? And is swearing falsely that I would not contact her, and swearing by my father’s mercy, an atonement for it?

The questioner is obliged to repent to Allah, because what he fell into falls under the general category of Zina (unlawful sexual intercourse) mentioned in the Hadith. As for swearing by other than Allah, it is forbidden, and there is no expiation for breaking such an oath because it is an invalid oath. Its expiation is through repentance, resolving not to repeat it, and uttering the word of Tawhid: "Lā ilāha illallāh (There is no god but Allah)."