In the estimation of GDP (using expenditure method), we focus only on expenditure by the residents of a country. Is this true?
No, this is not true. In the estimation of GDP, we include all expenditures on domestically produced goods both by the residents as well as non-residents of a country. This is why exports are added as they are expenditures on domestically produced goods, while imports are subtracted because money is spent on goods produced in foreign countries.