So... I think I understand this, but just to confirm: अपना means "his/her" when the noun belongs to the subject of the verb while उसका means "his/her" when the noun belongs to someone who is not the subject? (For people who know Scandinavian languages, this is like the "sin/sit" versus "hans/hennes," yeah? So a Norwegian translation of नेहा अपना सेब खाती है would be "Neha spiser sit eple" and नेहा उसका सेब खाती है would be "Neha spiser hennes eple"). Does this also apply for 1st and 2nd persons? For example one must say मैं अपना सेब खाता हूँ for "I eat my apple" (I assume that मैं मेरा सेब खाता हूं is incorrect...) and "you eat your apple" would have to be तू अपना सेब खाती है (I assume तू तेरा सेब खाती है would be wrong). Is this right?
I don't know any Nordic languages, so I cannot comment on that part (though it sounds very plausible).
In any case you described the distinction between अपना and उसका correctly. As I am also still learning Hindi I checked in two grammar books, to be sure. So you don't have to take just my word for it :D
Here is the rule from :
«The possessive adjective apnā, 'one's own' is substituted for the possessive forms of personal pronouns when the subject of the sentence possesses the object. apnā agrees with the noun it qualifies.»
So this applies - no matter what is the person/number of the subject. In  there are examples given with हम and मैं, in  one with मैं.
Apparently where अपना can be applied, it also has to be. Even if there is no distinction to be made about who the possessor is, e.g. when मैं is the subject.
( explicitly gives two examples marked as wrong,  says the substitution always takes place.)
However, it is possible to use both together, for emphasis. So this is the exact same structure as English «my own». Example from :
« [ यह मेरी अपनी गाड़ी है ]
- ye mērī apnī gāṛī hai
- This is my own car.»
(The Hindi/Devanagari text in square brackets is my rendering of the Urdu sentence from the book.)
 Schmidt, Ruth Laila. Urdu. An essential grammar. London: Routledge, 1999. p. 24f, para. 213.
 Hälsig, Margot. Grammatischer Leitfaden des Hindi. Leipzig: Verlag VEB Enzyklopädie, 1967. p. 71, para. 182.
I'd say this is because «her own» in English implies a strong emphasis (on posession) that is not present in the Hindi sentence.
It might help to think of अपना as «my/your/her/his/its/our/their own» to remember when it is applied. But it just replaces the 'normal' possessive pronoun (उसका in this case) when the possessor is the subject. So here it really just means the same as «her» in English.
And the other way around: If you want, you can even express the same emphasis in Hindi by using the literal translation of «her own»: «उसका अपना». (See also my answer to RaleighStarbuck)
Re-read my comments above. But basically you simply must get accustomed to thinking in Hindi- this language specifies possessives in a way that distinguishes between one performing an action on one's own object and on someone else's. In other words, a possessive adjective such as "his" changes depending on whether "he" coincides with the object or not. I know this might sound a bit abstract or confusing because this kind of distinction does not exist in other languages (I don't know what your native language is but I assume it does not do it), but see this example:
(1) He eats his apple. In English, this sentence is actually ambiguous- as it is, there is no way of knowing whether "he" eats an apple that belongs to him (i.e. the subject and the possessor are the same person) or if "he" eats an apple that belongs to some other male person (i.e. the subject and the possessor are NOT the same persons). In English, if we particularly wish to specify, we can add the word "own" after the possessive and say "He eats his own apple," but Hindi simply avoids this kind of ambiguity- it would be clear which meaning is intended because the equivalents for "his" versus "his own" would simply be different words (and likewise "her" versus "her own," "my" versus "my own,"etc.) The Hindi base word for "one's own" is अपना (of course, it must agree with the thing that is possessed. Therefore, to answer your question, "Neha eats your apple" would have to be one of the following (depending on the person who "your" refers to): नेहा तेरा सेब खाती है ("Nehā terā seb khātī hai)- intimate नेहा तुम्हारा सेब खाती है ("Nehā tumhārā seb khātī hai") - informal नेहा आपका सेब खाती है ("Nehā āpkā seb khātī hai")- formal
I hope that helps.
It helps somewhat. Not sure if I fully get it or not because the hints had 3 suggestions: her, your, and himself. Obviously 'Neha eats himself apple' doesn't work. But as far as 'your' goes (instead of 'her')...is 'your' sometimes spelled that way or is it just an incorrect hint that should be ignored?
Well, "Neha" is typically a female name (even if weren't, we know that it at least has to refer to a female in this particular case because the verb is in the feminine form- खाती ). Therefore, the English translation cannot be "Neha eats his apple" because that would imply that Neha eats an apple that belongs to someone else (i.e. the apple belongs to man). I believe what you're looking for is: नेहा उसका सेब खाती है However, note that this sentence can mean both "Neha eats his apple" OR "Neha eats her apple" since Hindi does not distinguish gender in possessives but in either translation it implies that the apple does not belong to Neha.