It's correct. But you must understand using 'want to' is asking what their 'desire', 'wish' or 'want' is. If you are not interested in what they desire to do, and you only want to know the duration of this stay, then it is much better to say: "How long are you going to stay there ?".

Yes that is correct... some alternatives: More polite (use would instead of want): How long would you like to stay there? (preference) How long do you want to spend there? For how long, would you like to be there? How long do you plan to stay there? How long are you planning to stay there? How long are you considering staying there?