If you include the article it means this apple. If you remove it it means this is an apple.

i don't know the rules, but "egy" is quite forgiving. you can put it before "alma", or "szék", or both of them and the sentence would still be correct.

The difference is that we don't know how many objects are in question. If you add an article you are defining that it is one apple/chair. If there is no article, it can be plural or singular, and if you want definite plural, you would add suffixes (Nem, ezek nem almák, hanem székek.)

If it's a definite noun (the apple, my car, John's friend), including a or az is necessary.

You use egy mostly only when you want to convey that it's exactly one item you're talking about. (Egy also means 'one' in English.) If the exact number is not of importance, you can leave it out.

You mostly use egy when you want to put focus on the fact that it's exactly one thing. But here that's not important. It could as well be multiple objects. Rather than the number, you're focusing on the nature of the object - is it apple or chair?