Not in this sentence. „Dziewczynki” can be either a genitive singular (like it is here) or a nominative plural. Here we have a classic example of an object, which normally is in accusative, changing to genitive when it's in negative sentence.
If we wished to change one girl into many girls we would get „Oni nie szanują tamtych dziewczynek”.
If I wanted to change one girl into many girls, my wife would kill me. :-)
Not that absolutely every negation needs Genitive - the ones that needed Accusative in positive sentences do.
You are not alone. All these cases. I have such a rough time with them. They're almost as bad as the pronouns that are implied, no wait, they're required, no, wait, their assumed, no, wait, no, you can't assume that!!!! Pronouns are the worst.
No, "egg" sounds like a very clear Polish 'e' to me.
I don't think there's a good equivalent of 'y' in English... Wikipedia suggests "short i as in bit", I guess I heard it being quite similar, but I can also imagine 'bit' being said with a Polish i...
In Polish schools (and possibly other countries' schools) little attention is paid to accurate pronunciation, so maybe that's why you "can imagine that". The short i is a good approximation, as long as you know what short i is supposed to sound like.
But be careful if you don't know English phonology well, because your language might have a sound called "short i" that sounds differently.