"Die" or "der" are definite articles and need in general a special context, which has been made clear before to be grammatically correct. In this sentence however, a new information is introduced, so we cannot refer to something already been mentioned. That's why we need "Das" and not "Der" or "Die".
No, as far as I know. In fact, they use additional particles like 'that over there' das da to help differenciate.
Can someone explain why you would use das for "that" and not something like "die" or "der"?
the demonstrative pronoun 'this' / 'that' just happens to be spelled like the article 'das'. Consider it a coincidence to help remember
Because this is nominative case, where the singular triplet is <dein, deine, dein>.
<deinen, deine, dein> belongs to accusative (that is, when it is direct object) as in I have your dog (ich habe deinen Hund).