how to determine the credibility of an online futures and stocks brokerage firm in United States of America.?

Answer 1

Without a name, it is impossible to answer. There are many very good and reputable brokers in the US who will accept UK clients. I use TD Ameritrade and they've always been excellent.

Most brokerage companies are quoted on the stock market, if you're thinking about using someone else one way to check them out would be to check the company records on somewhere like Yahoo Finance.