I am slightly confused with this construction. If ничего already means nothing, why is the нет needed afterwards. My instinct would be to consider that a double negative and to therefore mean something is on the plate.
Isn't a bowl миска? I know it's not a common word, though.
And is there even a difference between a dish and a plate? Are you sure тарелка и блюдо are not just synonyms?
It depends on the context. We rarely say "миска супа", rather "тарелка супа". "Миска" makes me think about cooking (and using bowls for storing chopped ingredients or mixing something) or about pets (кошкина миска = the cat's bowl).
There are also phrases:
глубокая тарелка = deep plate = bowl
мелкая тарелка = "shallow" or flat plate = plate
"On the plate there is nothing" is a word-for-word translation and it has the same meaning.
It's just an historical pronunciation thing. You just learn it and that's all, nothing logical behind it. Чего = Chevo, Него = Nevo, etc.