If we add up all the income while finding GDP of a country, is income counted multiple times?
There is no contradiction here. All these incomes are part of the GDP. In theory all incomes show the value of something. In your example this would be the value of the work done. Since all 10 government employees and the shopkeeper are working they all create value, the size of which is measured by adding up their salaries.

Here you can find an explanation of what GDP exactly is.