Why do we use the genitive in these lessons, how does is affect the meaning when it isn't in a sentence?
Somehow the grammar for numbers in Ukrainian involves Genetiv. Perceive it as, when we say "seventeen cars" we actually say "seventeen of cars". This is just the grammar, that's it.
Usually when you just name the thing ("a red scarf", "a big black car", "a huge pands"), you use the Nominative. The Nominative form of a comination of a number and a noun has certain rules.