I don't know about the significance of the black cap, but have wondered where the term "set your cap at him" came from. Anyone know?

As for scandalizing Mrs. Busby's nephew, earlier in the letter she had indicated that she was prevented from setting her cap at him (i.e., vying for his attentions and affections) "by his having a wife & ten Children." So after meeting him she stands corrected - he only has three. More of JA's wit.