Is it permissible to use the internet network of a non-Muslim neighbor if he has not placed a password for entry, knowing that the custom in his country requires placing a password?

Private internet is a right that cannot be used except with the owner's permission, whether the owner is a Muslim or a non-Muslim. Neighbors are more deserving of having their rights preserved. The fact that the line owner left the password does not imply permission, unless this is customarily understood. Anyone who wishes to use their neighbor's internet line must seek their permission.