I notice the recorded audio uses a rising tone for gì, despite it being written with a falling tone. Is that typical of this word?
No, the word "gì" still carries the huyền tone (falling tone). What you thought to be wrong was caused by the question intonation. Though Vietnamese is a tonal language, the intonation still plays an important part in speaking.
I notice there's no classifier here. Is it okay to omit the classifier in cases where it doesn't really add any information? Like if I point to an object, can I say "Đây là gì?" rather than "Đây là cái gì?"
Is this idiomatic, or is there a particular grammatical relationship in 'Bạn tên' that isn't apparent? I.e., I would expect 'Tên của bạn'.
The phrase "What is your name" is very common in Vietnamese, so we just format it "[Pronoun] tên gì?" or even just "tên gì?". "Tên (của) bạn (là) gì?" is also acceptable, and just sounds more "grammatical" but is otherwise a legal phrase. As a response, however, you will want to go with "tên (là) [name]", "[name]", or "tên (của) tôi (là) [name]".