Translation:Why is everyone talking about that girl?
This isn't a grammatical explanation, but to my (native AmE) ear, "Why is everyone talking about that girl" has a sense of there being a "buzz" about the girl -- lots of people are talking about her, they're talking about her a lot, and there's something unusual or special about they fact that they are.
"Why does everyone talk about that girl" is a perfectly fine sentence, but it doesn't have the same feel. Compare, for example, "Why does everyone talk about that girl SO MUCH" and "Why does everyone talk about that girl WHEN SHE'S NOT HERE?"
In the first sentence, "is talking" could easily be used as well, and it would convey the same sense of "buzz." But in the second, "is talking" would sound a little strange, unless the "talking about" is going on at the same time that someone is asking the question, and there is no "buzz factor."
Just one person's opinion, of course! :-)
While the meanings are very similar, they are not quite the same, and "speaking" sounds odd in this context. For more information, see https://dictionary.cambridge.org/grammar/british-grammar/speak-or-talk.
Všechny on its own, especially when used with a human verb such as speaking, means all the female persons. If clear from the context, it could also all other objects that are grammatically feminine (všechny televize, všechny stanice). Or it is used for masculine inanimate or in general for mixed-gender groups that do not include masculine animate and that are not neuter-only (všechna rádia).
Check the declension at https://en.wiktionary.org/wiki/v%C5%A1echen#Declension