If I divide a country nominal GDP per capita by 12, do I get the average monthly wage?

Afraid not.
GDP is the Gross Domestic Product - it is how much is produced by the economy as a whole. In an efficient economy, workers are paid a wage equal to their marginal product - the value they add to production. If you were in an economy where all production was returned to workers through wages, then what you are saying would be correct. However, this is not the case - other inputs also go into production, such as land and capital. Dividing GDP by 12 will give you the average monthly output of the economy, but only a fraction of that output goes to labor - approximately 2/3 in most developed economies, though I think (from memory) the US might be a little higher (3/4).