All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
The Prophet said: ‘‘If you hear the barking of a dog or the braying of a donkey by night, then seek refuge in Allaah from the devil because they see what you do not see.’’ [Ahmad and Abu Daawood].
It is clear that this narration is about dogs and donkeys, and that the legislated mention when one sees them is to seek refuge in Allaah from the devil. However, we are not aware of any evidence that this ruling includes cats as well, or that it is permissible to say ‘La hawla wa la quwwata illa billaah’ (There is no might nor power except with Allaah); in such matters one should restrict himself to the religious text.
As regards the cats becoming silent when one recites the mention of Allaah, then this may be a mere coincidence but it is likely that the mention of Allaah has an effect on them.
Finally, we do not know any Islamic evidence with regard to the last thing which you mentioned in the question.
Allaah Knows best.