Nope, it's not related in any way to English 'vegetable', „warzywo” comes from verb „warzyć” which used to mean 'to cook' – originally, only cooked vegetables were called that, uncooked vegetables were called „jarzyna” and even that was an effect of semantic shift. In earlier history, vegetables were distinguished between these that are planted in spring(„jare”) and these that are planted in autumn(„ozime”).
In a way, there is no "Z" in this word. Not as a sound. RZ = Ż (only a matter of orthography, but the same sound), and it corresponds to English ZH.
The number of situations when "RZ" has the sounds pronounced separately is super small: I can only think of "marznąć" (to feel cold, to be cold) and its derivatives, "mierzić" (to disgust, to sicken) and its forms, and "Tarzan" (which is pronounced the same way as in English).
It sounds like /'ВА-жы-во/ but Polish-speakers in this course have said that the accent in Polish words is almost always on the second-to-last syllable and that you can't go wrong pronouncing Polish that way. So in that case, it should be vazhívo