GDP = C + I + G + (X-M)
Gross Domestic Product is equal to the sum of Consumption Expenditures, Investment Expenditures, Government Expenditures and Net Exports. The Final Goods included in GDP, then, are goods in the hands of final users: Consumer Goods bought by consumers, Capital Goods bought by entrepreneurs and held by them as assets, Government Goods bought by government officials and Net Exports.
There are two omissions to note. First, GDP does not include the production of intermediate capital goods. So to produce automobiles, which would be included in GDP, iron must be mined, steel must be made, fenders must be formed, and so on. None of the production of these intermediate capital goods is included in GDP. Second, government transfer payments are not included. So, fiscal expenditures for F-18s are included in GDP while social security payments are not.
Yes, G adds dollar for dollar to GDP. So, when the federal government increased its annual fiscal expenditures from $2.9 trillion to $4.0 trillion in 2009, GDP was $1.1 trillion larger than otherwise.
The argument about including G in GDP is that since G are financed by coercive extractions, the prices paid for government goods do not, and cannot, reflect the value people place on them. So adding G to C and I is like adding apples and oranges. Even though they are both denominated in money, government expenditures and private expenditures do not have a common meaning.