Are there studies testing whether academia rewards researchers based on number of publications?
There's quite a bit of this sort of thing in economics. One early and now classic study, Katz 1973, performs a regression of salary on the number of books, articles, "excellent articles", and other variables associated with a faculty member, and finds what seems to me to be very low impact of publication on annual salary even in 1973 dollars: an article is worth $18 and an excellent article worth $102 in annual salary. 

A later study Diamond 1986, estimates the marginal value of a citation on lifetime salary: somewhere in the $50 to $1300 range (!!).  Note that is in economics where citations are somewhat stingy, back before people started gaming citation rates as much as they do today. 

Since there has been a huge volume of publications on the determinants of faculty salary within the economics literature. For some reason, this particular labor market has been of exceptional interest to academic economists.