Saying "Gin da." is grammatical and better than just saying "Gin.", but it sounds fairly blunt and unfriendly. In real life, basically no one will self-introduce like that. In anime, some prideful people and military personnel may speak in such a way.
In real conversations, da can be used in isolation to make a strong, accusatory or emergency statements where politeness does not matter. For example, "Jishin da!" ("Earthquake!"), "Omae wa baka da!" ("You're an idiot!"). Otherwise, usually da is omitted or used in combination with some sentence-end particles like "da-yo", "da-ne", etc.
Asking "Gin da?" is possible, but it's something like "Gin, you say?" It's a blunt guy's way of repeating someone's statement with surprise. One can even say "Mita da?" ("You say you saw it!?") I don't think it's very common even in anime, and it's very rare in modern conversations among ordinary Japanese people. Basically a question that ends with da is uncommon.