surely in any нет + genitive construction a plural form of the noun would be acceptable. I.e. here a natural translation would be 'there are no apples here.' In fact, I think it is way more natural in English to use this plural form. This keeps coming up with every such construction.
"There are no apples here" translated into Russian would use the genitive plural яблок. The likely reason the sentences are using genitive singular is because you haven't reached the genitive plural skill yet. So for now just continue to translate in the singular. This does sound a bit strange but it's not incorrect and I think it's mostly for the purpose of teaching you the genitive singular case.
The words 'да' and 'нет' do not always directly correspond to 'yes' or 'no' in English; English has this weird deficiency in the way to answer a negative question. If you were asked "aren't you coming to our place?" the answer could be: "yes, I'm coming", "yes, I'm not coming", "no, I'm coming" or "no, I'm not coming".
Russian bypasses this confusion by having 'да' always mean "I agree with what you say, you are right" and having 'нет' always mean "I disagree with what you say, you are wrong". The above example could go either two ways: да (you're right I'm not coming) or нет (you're wrong, I'm coming).
Thus, an exchange might go like this: "В этом месте нет яблока?" "Да, здесь нет яблока."
No, Russian is fond of double negatives. It's standard.
Я знаю, что ничего не знаю. - Сократ
That would probably be: яблока не здесь.
I think your version focuses on the "apple" as the the 'core object' of the situation while здесь нет яблока focuses on the "here" as the 'core object' of the situation.
"Yes, the apple is not here" should be accepted. The word "there" is nowhere to be found in the Russian phrase.
Why not "яблоки" instead of "яблока"? Because we have девочки, мамы... by that I mean the genitive form should end in и or ы.
Neuter words that end in «о» follow this pattern of declension.
nominative - яблоко, яблоки
accusative - яблоко, яблоки
genitive - яблока, яблок
dative - яблоку, яблокам
instrumental - яблоком, яблоками
prepositional - яблоке, яблоках