Learn Dutch: Heeft jouw vader zijn sleutels? - Does your father have his keys?

That rule only applies when the subject is jij or je (you), like in heb jij? (do you have?). In this sentence, the subject isn't "you", it's jouw vader (your father). Since "your father" is a "he" (third-person singular), we use the regular "he" form of the verb, which is heeft.

Can I use je instead of jouw here?

Yes! Je is the normal, unstressed word for "your". You would use jouw if you want to put special emphasis on the word, like asking "Does your father have his keys? (Because my father lost his!)"

Does the word zijn (his) change because sleutels is plural?

No, possessive words like mijn, jouw, zijn, and haar stay exactly the same whether the noun that follows is singular or plural. So it is zijn sleutel (his key) and zijn sleutels (his keys).

Why does the plural sleutels end in -s and not -en?

Dutch plurals usually end in -en, but words that end in an unstressed -el, -em, -en, or -er take an -s instead. Since sleutel ends in -el, its plural is sleutels.