Is ты necessary here? The subject seems to get dropped in many sentences and is fully implied in this case by хочешь. If it is, is there some rule of thumb for when the subject is necessary and when it is not?
Yes, in this case you can drop "ты". In such "propositional" questions you usually can omit it. There are a few other cases where you can like "пусть + инфинитив". But generally you need a subject even if it's implied by the verb's form.