Corrections in red.  Stylistic suggestions and variants in blue.

Nocturnal/Nighttime Cat/Feline Pranks. Is It Possible to Get the Better of It?

The existence of my family has changed since we took in a stray kitten some months ago. Like the majority of young cats, [1] our pet Grisha likes playing pranks/likes to play pranks.  

The cat sleeps most of the day, but when we go to bed, he begins pulling pranks/begins to pull pranks. Its/His [2] favourite game is to lightly touch its/his claws to [3] my sleeping child's/son's/daughter's feet. It is noteworthy that Grisha is smart enough not to/pretty smart not to touch my husband’s feet.

Well, I decided to impose sanctions against the cat, and we did not let him sleep after 6 p.m., in order to prevent his nocturnal/nightime pranks. Unfortunately, however, the policy was not successful.

Last night our pet made a terrible mess in the kitchen, [4] and pulled a mosquito net out of the oriel window. It only took him about half an hour.

Now we're going to do not let him sleep after 5 p.m.   Clearer variant:  Now we're not going to let him sleep after 5 p.m.

[1]  Variant:  Like the majority of kittens, our ...

 [2]  When you know an animal very well, you may refer to it as "he" or "she."  You can especially do that when you feel affection for the animal.

[3]  "To claw" suggests that the child's skin was broken.  I hope that's not happening.  If the cat is not breaking the skin, then it's better to say it the way I suggested.

[4]  To make a mess can simply mean to leave things in great disorder; to ruin or spoil something. However, in the context of a domestic animal, "to make a mess" may easily be interpreted to mean that the animal "went to the bathroom".