Translation:Please say your name.
You don't just say "please tell your name" it sounds weird in English, that is why it is wrong. If you want to use the word "tell" you have to have a direct person or object you're telling it to and it has to be notified in English. For instance, "please tell ME your name." Thats why it is not tell in this case, because it is not grammatically correct in English.
Sounds very rude by using あなた, a possible situation where this could be justifiably used would be a police officer questioning a suspect. Otherwise, 名前を教えて下さい" when politely asking for someone's name should used, and please, dont use with あなた also, unless you are an authority [ e.g. security personel or medical emergency person ] questioning someone...
I would just say "お名前はなんですか？" and then when you don't get it the first time say "もういっかい" to say "one more time" and if they are talking too fast say "もっとゆっくり"to say "more slowly". I would not over complicate what you are trying to say by repeating unnecessary information. You do not have to say all of the things you would say in English in Japanese. If you wanted to say "My name is Jas. I speak English" you would not say "私の名前はジャズです。私は英語を話せます。" it is just not practical. Nothing else is different either. It is not practical to say "あなたの名前はもう一回言ってください" it just becomes troublesome.
No, its not. I would not go up to your boss and say this, but it is not "rude" to say this.
People tend to avoid あなた but i have heard people say お前 to complete strangers before and お前 is considered to be even more rude. In my experience, お名前はなんですか? Is pretty okay to day no matter who you are talking to. Most Japanese people are just happy that you're speaking Japanese in the first place and do not really get ang pickier than that when it comes to foreigners speaking Japanese.
I misread お名前はなんですか for お前はなんですか (What are you?) and I think you did just the opposite. :D
お前（おまえ）is another word for "you," which is more impolite and straightforward than あなた, but I think you can use it to refer to any kids (esp. boys) when you're an adult, at least. It has nothing to do with お名前, which indeed is a polite word.
Another example of a very frustrating and inflexible answer. What is wrong with "Please tell me your name." This is a far more common statement in English than "Please state your name". The answers to each question need to be reviewed and other reasonable possibilities included!