> How many managed colonies of bees per square mile are there across the US?
Obviously, this is difficult to say for sure. The Census Bureau says in 2009 the US "managed colony count" was 2.46 million hives. However, they only count apiaries with 5 or more hives.
The 48 contiguous states in the US total 3.11 million square miles, so the average is about 1.26 managed hives per square mile. Clearly adding in an unknown number of hobbyist hives would increase that average somewhat.
FYI, there are 640 acres in a square mile.