Learn Hardcore Finnish: Avain voi kadota pian.

Actually, the word in question is voi, not vai. Voi is the third-person singular of the modal verb voida. Here it expresses possibility or likelihood (“may/might”) rather than ability. So Avain voi kadota pian means “The key may get lost soon.”

Why is kadota in its basic (dictionary) form after voi?

After modals like voi, the main verb appears in the first infinitive (basic form). Finnish never conjugates the main verb for tense or person when a modal precedes it. Instead of, say, kadottaa or kadotan, you keep kadota.

What’s the difference between kadota and kadottaa?

kadota is intransitive (“to get lost/disappear”). The subject (here avain) is what vanishes.
kadottaa is transitive (“to lose something” or “to make something disappear”). It would need a direct object in the partitive case (kadotan avaimen = “I lose the key”).
Since the key itself “gets lost,” we use the intransitive kadota.

What case is avain, and why is it not in another case?

Avain is in the nominative singular, the default form for the subject of a clause. Because it’s the thing performing (or undergoing) the action of the verb, it stays nominative.

What function does pian have, and could I move it elsewhere?

Pian is an adverb meaning “soon.” It can appear in different positions for emphasis:

End (neutral): Avain voi kadota pian.
Front (emphasis on soon): Pian avain voi kadota.
Middle (focus on the verb): Avain pian voi kadota.
All are grammatically correct; word order in Finnish is relatively flexible.

How would I say “The key will definitely get lost soon” instead of “may get lost”?

You can replace the modal voi with either:
1) A simple present verb form (certainty): Avain katoaa pian. (“The key will/does disappear soon.”)
2) A periphrastic future (tulee + first infinitive): Avain tulee katoamaan pian. (“The key is going to get lost soon.”)

What type of verb is kadota, and how does its conjugation look in the present tense?

Kadota is a Type I (–ta/–tä) verb. Present-tense conjugation:
1st sg. kadon
2nd sg. kadot
3rd sg. katoaa
1st pl. kadomme
2nd pl. kadotte
3rd pl. katoavat

Could we make this sentence passive or impersonal?

Yes, to make it impersonal/agentless you could say:

Impersonal active: Avain voi kadota pian. (unchanged)
Passive: Avain voi kadota pian doesn’t change much because Finnish passive often just marks that the agent is unspecified. A more “textbook” passive would be Avain voi kadota pian or Avain voi kadota pian. (Finnish passive is tricky here—usually you’d stick with the active intransitive.)