I saw this as two people serving up food.....One says "No, no rice for mum" .....The Owl didn't like that....heart lost !
This is an example of the problem with word for word translation. The "have" is implied by the whole у + subject in genitive + есть (есть often being omitted if the having isn't the point of the sentence). There isn't really a verb in Russian used the same way as "to have" is in English. It's the entire construction.
"у" translates to "at", "by" or "near". "У мамы есть рис." = "Mum has rice." (literally "By mum there is rice."). "У мамы нет риса." = "Mum does not have rice." ("By mum [there is] no rice"). Note that у is followed by genitive case, and in the negative нет is also followed by genitive.
So as far as I have understood, you don't need to use "есть" if the sentence is negative about having something
For example "у мамы есть риса" (mom has rice) "у маму нет риса" (doesn't have)
Is that how it works?
That's correct, when the phrase is negating the possession of something, you omit the есть
Probably because you used "nope" instead of "no". If I'm not mistaking, "nope" would be "нету" or "неа"