You can defo destroy shrines in your own region, i am just not so sure if its a lords only option or an option for every troop leader, i believe its the later.
I don't recall this being possible. Temples can be closed or raided by the lord, but I don't remember seeing such an option for shrines, which means that if you fail to destroy shrines before taking over a region, you likely will never be able to destroy it as far as I know.
