Regarding the removal, it used to be that once its within a certain distance of a habitable house you could cut a tree down.
Development plan standards might only be invoked if you seek permission before cutting it down, but you should get clarity on this.
A tree on the boundary can become the subject of dispute should you try to remove it.
You seldom see full tree removal nowadays, the tree trunk is cut low to the ground and the roots are left in the ground and ground down.