How do I use Stellar API to power my Universal Trading and Multicurrency Payment+Transacton Gateway? I am planning to build on top of Stellar a unified universal gateway+trading platform(all frequencies) for banks and financial institutions to power ultrafast international payment transfers in fiat/crypto currencies, automated equity and forex trading as well as interbank transactions involving trade financing instruments e.g. Conditional Lines of Credit, LCs and Bank Guarantees. I plan to write its low-latency backend in C++ and frontend will comprise of a PID server in combination with a web/JavaScript site as GUI.

The PID server monitors the PID’s of the strategies/transactions running on the server and executes the commands as they come. The server could consist of a webserver listening on port 8888 with a simple JSON interface, to make it super flexible.

How do I use Stellar API to accomplish my goal?
I'd start by reading the concepts here so you know what stellar will do for you vs what you have to do on your own: https://www.stellar.org/developers/learn/
It also has the API documentation.