"Look behind you."
Translation:अपने पीछे देखो।
I think of it like this: अपना (and related forms) means "X's self", where X is given by the subject. So we have आमिर अपने परिवार से प्यार करता है which breaks down to "Aamir loves X's family" with "Aamir" as X. In an imperative sentence, like this one, the implicit subject is "you", so we can really think of this sentence as being "You look behind yourself", which boils down to "You look behind X's self" with "you" as X. Therefore you use अपने instead of तू/तुम/आप
By looking at the form of the verb. Hear देखो is used, which corresponds to तुम. For तू it would be देख so just the stem of the verb. Finally, for आप one would use देखिये. Here you can read about all those forms:
According to the link above, there is also so called deferential imperative but it's rarely used.
Yes, अपना (here in the oblique form अपने) just means own or self:
It doesn't change according to the person. By the way, it is spelt अपने (with अ, not आप). आपने is the ergative (the form that marks the subject of a transitive verb in past tense).
अपना/अपनी/अपने are used to create a construct meaning something like "X's own", and doesn't change according to the subject that is doing the possessing.
अपनी किताब पढ़िए -- read your book (but not mine)
मैं अपने भाई से प्यार करता हूँ -- I love my brother
Keeping in mind the sentence literally translates to "look [at] your back":
Here, the subject is तुम and the object being possessed is पीछा ("back"). We force पीछा into the oblique case because it's the direct object of देखना (thus पीछे); अपना is governed by पीछे so it needs to be अपने.
No, because देखो is the imperative form for तुम, not आप. The command for आप would be देखिए. (देखिये is an alternate acceptable spelling.)
I'm actually baffled why Duolingo has completely neglected to teach any of the आप forms (at least on mobile) when आप is the safest way to address someone without risking offending them for being too informal.