How many manuscripts per day are sent to leading journals?
According to Nature's website, they received 10,952 submissions last year, which works out to an average of almost exactly 30 papers per calendar day. Given that there are 26 subject area editors according to their editorial masthead, it would appear that the average editor gets about 8 papers per week, or one or two per day. 

However, their caseload is probably much bigger, since they also have to deal with all of their other "open" papers as well—so an editor might have something like 60-100 active papers to deal with at any one time (ballpark estimate, assuming a two- to three-month life cycle—although I suspect it's actually much longer than that if multiple rounds of reviews are required).