Cat behavior question

I am writing this for a friend. I will try to forward the replies to her.

She has a 3 year old neutered male cat which was born to a feral mother. She
brought the cat inside at 6 months and neutered him. He is average size, white
with grey spots and a grey tabby tail, very pretty, shorter fur. He likes to
sleep on the bottom of her bed and will lay next to her on the couch. He likes
to sit and play with the dog. He's not a lap cat and isn't too fond of being
picked up. The woman has two boys, 7 and 4. The cat now approaches them and is
somewhat friendly.

Soon after the cat was neutered he started escaping from the house and spraying
the neighbors front door. Maybe a new male cat entered his territory? He would
rip out the screens if a window was left open. He would slip out any door that
was opened. He would always come home. The neighbor is complaining about the
spraying. That is the main complaint. She said she can't keep him in the house
as he escapes. He was given amitriptyline 10 mg once a day for two months to
keep him from escaping. He was a zombie. She reduced the dosage to half and it
had no effect.

She needs some ideas about how to keep him from escaping and spraying the
neighbors door. If she can't find a solution, she feels she may be forced to
rehome him because of the complaints. She would like to keep him. Anyone with
any suggestions? Does he need more attention and play time? Feliway? More
secure window screens made out of thicker metal? Any suggestion no matter how
wild would be greatly appreciated.