Learn Finnish: Pihditkö sinä tarvitset vai pillin? - Is it the tongs that you need, or the pipe?

This is contrastive focus. By moving the object (pihdit) to the front and attaching the question clitic -kö, you are specifically asking if the tongs are what is needed. A standard question would start with the verb: Tarvitsetko sinä pihdit? (Do you need tongs?).

Both pihdit and pillin are objects of tarvitset, so why does pihdit end in -t but pillin ends in -n?

They are both in the accusative case (the total object case), but they follow different rules because of their grammatical number. Pilli (pipe/whistle/straw) is singular, so its accusative form takes the -n ending. Pihdit (tongs) is a plural-only word, and the plural accusative always looks exactly like the plural nominative, ending in -t.

Why does the question clitic on pihdit use -kö instead of -ko?

Vowel harmony. The vowels in pihdit are both i, which is a neutral front vowel. When a word only contains neutral vowels (i or e), it takes the front-vowel version of grammatical endings, so it becomes -kö.

Could I use tai instead of vai for "or" in this sentence?