Yes, governments should use open source software, because that would make it easier to reach people with government information. Other people could use open-source software in order to improve on what the government has done. It'll also be helpful to critique what the government does and make recommendations for improvement. It would make the information more free.
Absolutely, the government should be using open source software, as much and as often as possible. The government should check open source software to make sure that it could do the work that the government needs it to do and find something else if the open source software doesn't fill it's needs but the government could save the taxpayer money by using open source software and encouraging others to do the same.