I think both. It can certainly mean “I have no hair”. I guess if you wanted to specify “hairs” you might say “Körperhaare”, otherwise context would be enough.
“Haar” is neutral, but this is the plural form “Haare”, this is why you use “keine”.
That doesn't sound super natural in English - we usually use "hair" as a mass noun.
The translation is correct, if it doesn't accept it, it simply means it isn't in the system (yet). You can report it as a correct translation the next time you find this exercise.
I've added it. It was a new sentence lacking alternatives - that happens sometimes! I can only echo the advice to send reports for answers that should be accepted. They find their way into the system eventually, if they are correct.