]-- In P&P2, when Mrs. Bennet introduces her daughters to Bingley at the assembly, why does she refer to her second daughter as "Elizabeth", but her fourth daughter as "Kitty"? Shouldn't she have introduced Kitty as "Catherine"? Or if she's determined to use nicknames, why not call Elizabeth "Lizzy"?

Perhaps it is to emphasize how little respect Kitty got from her family, esp. her mother in general. First, it is the coughing, then Mrs. Bennet asks her to give her bonnet to Lydia, now she introduces her using her nickname.
Maybe she thinks Kitty won't amount to much marriageably- she has neither Jane's good looks, nor Lydia's 'good-humour', nor Mary's 'accomplishments'!