From the article and the pronoun. τα δικά μας is for plural. The singular would be "Το δικό μας σάντουιτς." ^.^
Yes, because technically, it's not correct. There is a difference between our and our own. "Our sandwiches would be just "Τα σάντουιτς μας" in Greek. The word δικά gives emphasis, just like own* does in English. ^.^
Yes, I don't get why. (I am an English teacher and writer of English course books.)
When someon says "το δικό μου..." in Greek, the emphasis is given on the actual possession.
To make it a bit more clear:
Someone asks you what are you doing right now. You'd reply with "I'm eating my sandwich - Τρώω το σάντουιτς μου."
Someone accuses you of eating their sandwich. You'd reply with "I'm eating my own sandwich - Τρώω το δικό μου σάντουιτς."
No, how? It doesn't end in one of the standard declension endings :)
A fair number of loanwords are not inflectible because they don't fit the "shape" of Greek words.
(Old loanwords tended to be assimilated, often by adding -ι, in order to make them inflectible. But modern ones are often borrowed as-is.)