I think it was explicit in the novelization of previous chapter that he did leave her on read and so she knows he's lying about the phone running out of battery
Not really, her ploy was just to look good to the guy by offering her support to him rather than blaming someone else for the leak. MC becoming a hikki seemed to have happened for many reasons across the school years
It's implied that they divorced after their daughter died, which is actually a pretty realistic scenario when two parents handle the loss very differently
Mr. Tiggy was going to die from his injuries otherwise
Also he didn't become human but rather a bakeneko who could still transform back to a cat, I think that what was he trying to do was to transfer his being a bakeneko to him instead