Why did Lot's wife turn round, in spite of the warning not to?
Targum Yonasan: To see what was happening to her father's family, since she was a Sedomian.
Behind whom did she peer?
Rashi and Ramban: Behind Lot, who had gone to the back, to urge the members of his household.
Targum Yonasan: Behind the angel.
Why did she turn specifically into a pillar of salt?
Rashi: Because she refused to give a little salt to her visitors on the previous night. 1
Targum Yonasan: Because she went round to all her neighbors that night, asking for salt for her visitors, in order to publicize the fact that guests had arrived, and prompting the people to come and take action (which they promptly did - Na'ar Yonasan).
Interestingly, Chazal equate salt with Chesed (See Kesuvos, 66b [of which S'dom and Lot's wife were the antithesis]). Perhaps that is because salt is in itself unpleasant to eat, yet it renders most other foods edible and tasty.