Learn Hardcore Finnish: Minulla on avain mukana.

Mukana: an adverb meaning “ <a class="border-b border-dotted border-blue-400 dark:border-blue-500 cursor-pointer text-inherit no-underline" href="/grammar/finnish/postpositions/kanssa">with (someone),” indicating accompaniment or physical presence.
Mukaan: an adverb often used to express direction or the act of taking something along (e.g., ottaa mukaan “to take along”).

In Finnish, possession is not expressed with “to be” plus the possessor as the subject (like Minä olen avain). Instead, you put the possessor in the adessive case (minulla) and use on (“is”). Literally Minulla on avain means “At me is a key,” which idiomatically becomes “I have a key.”

What case is minulla and what does it indicate?

Minulla is the pronoun minä in the adessive case (ending -lla/-llä). The adessive expresses “at” or “on” someone. In possession sentences, it marks the person who has something.

Which word is the grammatical subject in Minulla on avain mukana, and how does that affect the verb?

The subject is avain (“key”). Since avain is third-person singular, the verb is on (“is”). The possessor (minulla) is not the subject, so you don’t say olen.

Why is avain in the nominative case, not partitive?

In this existential/possessive construction, the possessed noun acts as the grammatical subject and is therefore in the nominative. You would only use partitive if the verb required it (e.g., after negation).