Learn Finnish: Jäitkö sinä laivaan? - Did you stay on the ship?

This is the core rule for the verb jäädä (to stay/remain). In English, we stay in or on a place, but Finnish treats this verb as movement into a location where the movement then stops. Therefore, jäädä always requires the illative (into) or allative (onto) case.

How exactly is the word jäitkö built?

It starts with the verb jäädä (to stay). The past tense stem is jäi-, to which we add the personal ending -t for 'you' (giving jäit, you stayed). Finally, the question particle -kö is attached to turn it into a yes/no question.

Could I drop the word sinä and just say Jäitkö laivaan?

Yes, absolutely! Because the verb ending -t in jäitkö already tells us the subject is 'you', the pronoun sinä is optional. Including it just adds a little extra emphasis, like asking 'Did you stay on the ship?'

I've seen ships take the outer location cases like laivalla (on the ship). Why is it laivaan here instead of laivalle?