Duo accepted "Planet Earth is a small star." Grammatically correct, apparently, but pretty confusing.
But it fits the original sentence which is not, strictly speaking, correct, because כוכב is "star", while Earth is a planet.
I am a student, but according to the lesson's notes that is exactly what it says. That it means both star and planet, whereas כוכב לכת only means planet
Yes. It can also refer generally to many celestial objects. For example, a comet is כוכב שביט.