Learn Finnish: Tämä on se uusi ammatti, joksi minä vihdoin valmistuin. - This is the new profession as which I finally graduated.

joksi is the translative case of joka. It is triggered by the verb valmistua (to graduate). When you graduate to become a certain profession, Finnish treats this as a change of state, which always requires the translative ending -ksi.

Could we use the essive case (jona) here instead of the translative (joksi)?

No, because the essive case (jona) describes a continuous state of being, like working as a teacher. Since graduating implies a transition or becoming something new, you must use the translative case (joksi) to show that change.

I thought se meant 'it' or 'that'. Why does the English translation say 'the new profession'?

While Finnish doesn't have official articles like 'a' or 'the', speakers often use se to act like a definite article. It highlights that this is the specific profession you were likely just discussing, rather than just any new profession.

Can I drop the pronoun minä from joksi minä vihdoin valmistuin?

Yes, you can completely drop minä and just say joksi vihdoin valmistuin. The -in ending on the verb already tells us the subject is 'I'. Including minä here just adds a little bit of personal emphasis.