From DL's Tips & Notes:
In Hindi, sentences like “I eat my apple” seem too redundant (मैं मेरा सेब खाता हूँ). So, a class of three different words are used to replace possessive pronouns when their (pro)noun has been mentioned before.
अपना is used for masculine singular objects, अपनी for female singular or plural objects, and अपने for masculine plural objects. Example:
- मैं अपना सेब खाता हूँ। – I eat my apple.
- तुम अपनी किताब पढ़ती हो। - You read your book.
- उसके बच्चे अपने केले खाते हैं। - His children eat their bananas.
Sometimes, these words can also be used when their noun or pronoun is obvious from the context. For example, when a kid comes up to you, you might ask them “अपना नाम बताओ” – Tell me your name (बताओ – imperative form of tell), where it is obvious that अपना means your (Notice that me is not required in such a sentence).
These words also take up the oblique case. For example, वह अपने घर से आती है – She comes from her home (आना - to come; अपना घर changes to अपने घर because of से - from).
I can see a very good reason for using अपना if the subject is a 3rd person. In English, a sentence like "He gives you his banana" is not 100% clear: does he give you his own banana or someone else's? Using अपना in the Hindi translation of this sentence would make it clear that the banana is his own.
However, I must say I don't see any reason for using अपना with the 1st and 2nd person subjects... Maybe the 3rd person case was extended to the other persons for whatever reason, despite the use of अपना is not necessary then?
Apnaa replaces the possessive pronoun (which could be teraa as you’ve said) when it refers to the same person as the subject of the sentence. So for example, तू अपना केला का रहा है। : YOU are eating YOUR banana. But if the subject and owner of the noun being possessed are not the same, the regular possessive pronouns apply, as such: वह तेरा केला का रहा है।
tumhe & tujhe mean the same as tum ko & tujh ko.
It's just a different (less formal/शुद्ध I think) way of saying the dative, than indirect+को, a variant form that आप doesn't have.
('Why' it doesn't have an alternative form I don't know, probably 'just doesn't', although a sort-of answer could be that प + ह never form a conjunct without the implicit vowel, would be hard to make that sound different than फ even if they did. So a hypothetical variant form of the dative would be different than 'aaphe', and so if it existed the question would just be 'why is it not indirect+(h)e like the others' instead of 'why does it not exist'.)