"Is it permissible for me to kill the neighbor's annoying rooster secretly, while compensating them for its price, if they do not respond to a request to stop its harm? "

Praise be to Allah, Islam encourages kindness to the neighbor and considers harming them as one of the major sins. So whoever believes in Allah and the Last Day should honor their neighbor and not harm them. Neighborly rights are mutual, so each of them should be kind to the other and refrain from causing harm. Neighbors should not disturb each other; problems can be solved first by advice and patience, and if that does not work, then by complaints to the responsible authorities. It is also not permissible to infringe on a neighbor's property as it may lead to worse consequences.