These are all excellent questions. Let me take them in order. According to statistics, Australia, Chile, Belgium, and Brazil always ranks as countries that have the greatest turnout for voting. It is sad to say that the United States has one of the lowest number of people voting.
Part of the reason why these countries (mentioned above) have greater voter turnout rates is due to the fact that voting is compulsory in these nations. So, if the United States wants to increase voting, then they could consider making voting compulsory or to charge a little fine for not voting. This should get people to the polls. However, this is not the whole story, because some countries have very high voting rates that do not have a compulsory system in place, such as Italy and Germany.
One of the reasons for the relatively high voter rates in these countries is undoubtedly due to the homogeneous nature of these countries. To have a mixed society such as the United States, it will be much harder to get people to the polls due to language and cultural barriers.
Finally, in terms of the final question, it is hard to say. Correlations are notoriously difficult to establish. In view of this, I tend to believe that while it is always a good thing to have more people vote, this factor alone does not seem to make a huge difference in terms of how well a government is run.