Why is "pokazywać" followed by the genitive case here?
Elsewhere it is followed by the accusative case (e.g. "My pokazujemy wasze zwierzęta", "Pokazujesz zieloną książkę")
"Pomidor" is an exception – it's treated often as animate noun. So what you see is still accusative, which happens to be identical to genitive as well. Declinating it as inanimate might be considered as a sign of hypercorrectness. This specific exception was mentioned in one of the tips for exercises.
It should be acc. But it is often mistaken by Poles so it is accepted colloquially. Anyway, report it.
Very rare that you would actually say this in English. Maybe at a village fete. Maybe.
So tomatoes are animate in polish? How come? Do tomatoes have a living quality that other fruit and veg dont have?
actually tomato is not alone. Banana, potato and cucumber are other that come to my mind. Food related nouns have many exceptions like that.
"czerwona" (or "czerwoną" here) is a feminine form, "pomidor" is masculine.