Is Тебе нужно помыть овощи. also possible? Is there a difference between нужно and надо. And where does должен fit in?
There is no difference in these words (надо and нужно). Your choice should be accepted. Press the report button next time.
A word "до́лжен" comes from a noun "долг" = "debt" or "duty".
A word "ну́жно" comes from a noun "нужда́" = "need" or "necessity".
So, the meaning of your choice woud be a little bit another, but, however, in this case they can be interchangeable.
Given that помыть is a perfective verb, should "you will need to wash the vegetables" be an acceptable answer here?
you will need to wash=тебе надо будет помыть. Perfectiveness doesn't imply a future tense. Here it means one-time action.
According to wiktionary, you use будет мыть but not будет помыть. будет is not listed in the conjugation tables of perfective verbs.
"тебе надо будет помыть" = "тебе будет надо помыть", but the latter sounds awkward for some reason. It would sound a bit better if we replace "надо" with any synonym which more resembles an adjective: "тебе будет необходимо помыть". Also I can say "ты будешь должен помыть" (will be ought to) and "ты должен будешь помыть" and so on. Those are awkward options but they would be understood as the logic stands. So this "будет" relates to "necessary"/"нужно".
i thought perfective doesnt exist in present tense. it must be future or past doesnt it?
So is there a difference between помыть and мыть? Is it just perfective/imperfective?