Slate on Guns
1) Your article seems to assume that each background check means that just one gun is sold. In fact, any number of guns might have been purchased with that sale. It is quite possible that the number of checks could fall and the number of guns sold increased. The reverse is also true. This seems like a pretty important point to at least acknowledge given your piece. My guess that in the current period gun sales are rising by more than the number of background checks.
2) (Nota bene: These days, 1.6 percent of gun applications are denied each year, translating into no gun sale.)
-- initially denied. This is not to say that they were permanently denied. It is very difficult to figure out how many guns that translates into for multiple reasons. Some may be denied a second time. You also don't know how many guns were involved in each sale.
3) National Opinion Research Center at the University of Chicago. NORC found that gun ownership in the United States has been falling since 1977 (PDF, Page 11),
-- The NORC survey is basically the only survey that I know of that has shown this type of drop over time. That by itself should be enough, but Tom Smith who ran the survey for many years had strong views on the gun issue and the survey was also received funding by the Joyce foundation.
A reader who posted a comment below brought up another excellent point that I am kicking myself for not mentioning: in some states permit holders are exempt from having to get a background check done and as the number of permits have gone up this would cause the number of background checks to fall. The numbers also don't include firearm transfers even in those states where a National Instant Criminal Background Check System (NICS) check is required for such a transfer.
Other NICS info is available here.