[for your question] 「あなたは？yours?/あなたのお名前は？」 would work fine. When you ask as the first, it should be like 「あなたのお名前は何ですか？What is your name?」 「お名前を聞いても良いですか？May I ask your name?」 Sorry, it came the last after the super long sentence...

As you have confirmed, in Japan it is rude to use "あなた" in a conversation. However, it is okay for the first conversation. After you know each other's names, you can use them, and often you can talk without them. ex：あなたの趣味は何ですか? It is more natural to simply say =「趣味は何ですか?」 Japanese people often speak without using the subject（あなた・名前）.

I don't think あなた is considered rude, basically. Normally it's rather thought of as the polite expression of "You". The famous one as a rude meaning word is あんた. This word is what people use when they get angry at what the others did/said to them, and attempt to tell it to the others. However, there's a chance that people would think あなた as a kind of rude expression but it's a very limited and technical situation in a way. It can be a rude expression when someone dares to use あなた even though they know your name and they have been used the name to call you so far. It is like, using too much polite expression that's not appropriate for the occasion, it can be an indirect disdain expression. (You can tell it by the fact that their way of calling you is suddenly changed, and from their attitude/their facial expression.) Why Japanese people take it badly in a certain situation like that even though あなた is basically a polite expression. It is related to the Japanese people's way of thinking. Japanese people divide people's impressional categories within their minds. As an ally(people who belong to you and your territory/field. Like family, friends, coworkers who're close to you, people who’re your side) or the others(who're not, like strangers to you, people who have different opinions against you= enemy-like existence to you) It's called inside ウチ and outside ソト concept/thinking. So this explains the situation above. あなた is basically thought a polite way of calling 'you', and it can be used when you don't know their name. (let's call it usage B) Even though they have been calling you by your name, changing a way of calling people into あなた on purpose and all in sudden in a conversation, it means that they came to want to take a mental distance from you.

初対面なら「あなた」と言ってもいいです。 ですが、日本人はあまり使いません。 相手の名前を知らない場合は、この様に相手を呼びます。 （名前を知らなくていい場合も多いです。） マスター(Masta^. Use for a Tea shop master) 大将(Taisho^. Use for the chef of a Japanese style pub) 通訳さん (Mr./Ms. Translator) ガイドさん (Mean:Mr./Ms. Guide) おばちゃん(Mean:Aunt) 大家さん(Mean:House owner) あなたが自己紹介する場合は、「私は、〇〇と言います。よろしくお願いします」 と、まず言います。そしたら、あなたが名前を覚える必要がある人は、相手が自分から名前を言うでしょう。 あなたが、相手をなんと呼べばいいか分からない時は、「なんとお呼びすれば、よろしいですか？」と聞いて下さい。 そしたら、相手が「”おばちゃん”と呼んだら、いいよ」「大家さんと呼んだら、いいよ」と答えてくれます。 In the first time, you meet them, you can say "Anata no namae wa nandesuka?" In this case, you can use Anata(But Note, Japanese does not use so often). If you want to speak more more polite, it is better to say "Nanto oyobi sureba yoroshi^ desuka?" Sometimes you do not need to know his/her name. They will answer "Obachan to yonde to yondara i^yo(Please call me Obachan(Aunt) )" 勉強、頑張ってね。