It depends on how the state laws are written. But in Nebraska the law excludes police from its texting and driving laws.
If the cops check your phone records which I highly doubt unless you give them a reason to.
The cops can read your texts
It's not illegal to text them but if the text has anything to do regarding sex or threats etc it is illegal.
no, it is not illegal to text and drive although it is in 25 different states.
no it isn't
No, it is not illegal, but it is extremely annoying.
all of them
No, it is no more illegal for a teacher to text a student than to email a student. It would be inappropriate for such to happen without the prior knowledge of the parents of the student, but not illegal.
Harrassment by phone is illegal, so yes, it is illegal.