how does the rate of gold was fixed ? Is this any relationship with the stock exchange market?

Answer 1

The value of gold in the US is not "fixed." The value is determined strictly by market forces: what willing buyers pay to willing sellers.
There is no direct connection between gold values and stock market values.
Some companies who mine and process gold are publicly traded, but the value of the stock is based on the ability of the company to earn money. The ability to earn money in the business may be affected by the market value of gold.

Answer 2

Seperate market. No direct relationship. Markets fix almost all issues if left alone.