What does the distribution of birth order look like? That is, of the
people in either the U.S., Western countries, or the world, what
percentage are only children, what percentage are firstborn, what
percentage are 2nd-born, etc.? A bonus will be given for specific
information discussing how this has changed over time.

perhaps this will be a good start for your inquiry??

http://www.isr.bucknell.edu/Doing_Research/Statistics_Guide/Statistics_about_Children.html
http://www.childstats.gov/contacts.asp

This may be of interest to you:

"The proportion of firstborns in the population -- and therefore the
decision about whether the proportion in a given sample is higher or
lower than expected -- varies over time, due to changes in average
family size. In U.S. and Canadian populations, the proportion of
firstborns was 27% in 1960, 39% in 1970, and 44% in 1984 (Somit et
al., 1996)."

http://xchar.home.att.net/tna/birth-order/believe.htm