How do I know if an output is unspent when looking through the raw transaction data of a transaction? When looking through the data represented in JSON, how do I know if a particular output has been spent as input in another transaction?
You don't. To know if an output was spent you must check the blockchain, the information is not encoded in the transaction itself.

In Bitoin, you don't need to scan the entire blockchain, though. There's a database maintained by nodes, wallets and other software called unspent transaction output, or UTXO. Other coins may be different.