Not that common, no.

Most native speakers would know and understand the word. It is not common. It is somewhat formal. "Destitute" means "very poor." The words we would be most likely to use would be "poor" and "living in poverty." We would used "destitute" and "destitution" as very strong words meaning extreme poverty. "When I was a kid, my family was poor, but we weren't really destitute. We always had enough to eat." "Homeless" is a related word. It is common. It doesn't mean the same thing as "destitute" because you could be destitute and still have a place to live.

Yes. It is a word anyone fluent would know. Not every conversation will use this word, sort of depends on what sort of thing you discuss, daily, but it is just another word in everyone's vocabulary.