How do win streak bonuses work in Hearthstone?

Do win streaks stop at a certain level?

Yes, you no longer get win streak bonuses at ranks 5 and higher.

Are there different requirements at different ranks for win streaks?

No, it's always three wins at all ranks where win streaks exist.

There are no bonus stars at rank 5 and above. Otherwise you could reach legendary rank with a win rate lower than 50%.