Нужен - male form
Нужна - female form
Нужно - neutral form
Нужны - plural form
Нужно is used with nominative version of nouns. In this case it plural form of овощ which is овощи.
Мне нужен отпуск (I need a holiday)
Нам нужна собака (We need a dog)
Тебе нужно время (You need time)
Им нужны книги (They need books)
The long form is нужный
That's what you have to use for words in cases other than nominative, with appropriate changes to the ending, depending on case, gender and number
Generally we use "which" when there is a limited number of choices otherwise use "what". Without a context we don't know.
E.g. we're in the kitchen. What vegetables do you need? Now, we're in the market faced with a selection of vegetables. Which vegetables do you need?
Yes that would be more correct, but in English many people use "what" to mean "which". It's colloquial but not truly correct
"Which" applies to a limited collection of things (two or more), and asks that some of them (or even all of them) be selected, e.g., "I have a peach, a pear, and an apple. Which do you want?"
The apple The peach and the pear, the peach and the apple, the pear and the apple All of them (for a choice of two possible items: "both of them"). None of them - I want an orange.
To limit the selection: "Which two do you want?" "Which one do you want?"
"What" is open-ended and non-specific - it can mean one or more things, depending on context. "I'm going to the store. What do you want me to get you while I'm there?"
A peach, a pear, and an orange.
You would not ask, "I'm going to the store. Which do you want me to get you there?" because the selection is open-ended.
Short-form adjective, whose plural ending is -ы (or -и if the spelling rules apply). See the post by UtkuAkdemir (above).
Yes. A clue to that is the use of the short-form adjective нужны, as short-form adjectives apply only to nominative nouns.
тебе is dative case, I believe.
I think that a literal translation of the exercise is something like: "Which needed vegetables for you?" or "Which vegetables (are) needed by you?"
I present the first inelegant literal translation because I have a grammar question: Can short-form adjectives be used as predicate adjectives, or do they have to be attached to a noun (as attributive adjectives)? If they can't be predicate adjectives, then the first translation is the only viable literal translation, because the second literal translation uses "needed" as a predicate adjective.
The reason that's relevant is because, if short-form adjectives can't be predicates, then you'd have to use the long-form adjective, or one of the adverbs that can be used as predicate adjectives (e.g., больше).