score:13
To answer your first question, yes, the total Christian population is growing, by about 1.5% per year. This number comes from one of several excellent studies of Christian populations, a 2013 report published by the Center for the Study of Global Christianity, at Gordon-Conwell Theological Seminary. (full report)
This particular study focuses on the growth rate of Christian populations, and provides details by country, so it's particularly suitable for the questions you pose. It compares the number of Christians in 1970 to the number of Christians in 2010 and the projected number in 2020, and provides rates of growth, for each country. For example, here's a table showing countries with the most Christians:
And here's the growth data for Central African nations (similar tables are available for other groups of nations):
It also provides some growth data by faith tradition and continent:
There are other excellent reports out there, which focus on different things. The 2011 report published by the Pew Research Center (full report) focuses less on the growth rate but has some helpful graphics showing the shifts in global Christianity. For example, a comparison of regional distribution between 1910 and 2010:
And a map of Christianity by country in 2010:
Note that this is only a small sampling of the data available in these two reports; the full versions go into a lot of detail about individual countries and faith traditions.