These are all great questions but...
With the exception of the first question, they didn't know these details yet. They didn't have phone records, so they couldn't ask question 2, they didn't interview Tony yet so they couldn't ask question 4, with question 3, they didn't interview friends or Tony yet to know that she just started going to Fusian.
Agree, I have to give high kudo praises to LE for going along with her lies for as long as they did in the beginning. LE has heard every story under the sun and their BS radar went off immediately after listening to the last 911 call with KC and the 911 operator. Unfortunately with KC lying the way she did (has), it takes time to get facts together. They are still doing just that.
We all should be thankful that KC didn't lawyer up immediately and she did spew all her lies because if she had, she might still be out on bond and Caylee might not have been found as soon as she was.