英语里一般是不会出现“who who”的,要尽量避免使用两个相同的引导词.所以最好说“who that”.
你的第一个句子的提问应该说“Who is from America that likes music?”或者“Who likes music that is from America?（更好）”
而第二个句子应该问“Who is Tim talking to that is writing an e-mail?”或者“Who is writing an e-mail that Tim is talking to?”