"Tell your name."
Translation:अपना नाम बताओ।
It has to do with the subject being the doer to something that belongs to him or herself. I see my brother. She talks to her mother. They give their teacher a gift. In these cases we us a form of apna, because the thing following the word apna is a thing that refers to the subject here. So, in your question, "tera nam batao" it is wrong because the person has to say his or her own name. When we ask "what is your name?" we don't use apna because the verb is "is", and doesn't have as a subject the person "YOU". Hope this helps.
No, for a couple reasons:
I think you're confusing अपना (reflexive possessive, basically "your own") with आपका (your). You need अपना/अपने/अपनी when referring to something the listener possesses as opposed to the listener themselves.
The command for आप is बताइए; the command for तुम is बताओ. Colloquially you may hear use of the तुम forms for आप but it's nonstandard.