I do not know of any other way of transliteration— Дженни seems the only possible option. If you were wondering, Russians, of course, speak about foreigners and read fiction. We are familiar with popular foreign names. Using Джон seemed like a laughable choice, so I went for Дженни because I needed a girl's name anyway.
Note that native girl names fit into the usual declension pattern for typical feminine nouns and have different forms for different cases.
This is clearly a question, but the intonation sounds like a statement. I was under the impression that in Russian, you communicate that something is a question primarily by changing the inflection or intonation.
Is this a bug / mistake / flaw in the audio, that it doesn't correctly do the question intonation, or am I missing something?
The intonation sounds like a question here. Not the expected question, though: it places emphasis on "tennis" rather than "play".
A statement would have the pitch bubble pretty uniformly and falling at the end. A yes/no question has an intonation peak on the stressesed syllable of the question "core". A wh-word question will have a weak peak on the interrogative word and maybe on the core; sometimes it has no peaks.