"Verdura" is like "frutta; it signifies a collective as well as singular. When you pluralize "verdura," you indicate plurality, diversity of things. The sentence is asking the quantity of the general vegetables together being cooked. If it was "quante verdure," then the question is asking "how many kinds of vegetables."
Because vegetable is a singular noun. You can only use 'much' with nouns that can be both singular and plural like water, fish or fruit- or it's an emotion, you can use much with that too. One rarely uses the single noun vegetable unless they're describing a particular one. Like "How much of this vegetable does he cook?" Otherwise it's always "How many vegetables," and it would be implied that they meant how many types of vegetables, otherwise they would have asked about the particular one they were referring to with regards to quantity.
much is used with quantities such as water that we don't count. We cannot say "How many water? that would be wrong because we cannot count: "1 water, 2 water..." We can say "1 vegetable, 2 vegetables, 3 vegetables ..." or "3 carrots, 1 onion, 2 potatoes, 1 turnip, 20 green beans..." which means we use "How many vegetables does he cook?", but like you said you can say "How much of this vegetable does he cook?" or "How many cups of vegetable does he cook?" In Italian, it is different "Quanta" works for both "how much" and "how many". It is only when translating from Italian to English that you have to be careful.
I tried that and it came up incorrect, I didn't think "does" was implied but it said "how many vegetables does he cook".
I believe it should be accepted, but i could be wrong.
The simple fact is languages don't have an initial intent of being translated; they are meant to be used in their own context, so it's sometimes awkward to really make sense of it in another language's context...
There's no sense complaining that this FREE (it's quite powerful considering that) language learning tool isn't omnipotent just leave feedback, and it can improve that's all.