You definitely need to check prior to tree removal as you will want to see if there are any regulations within your council or state which regulate the removal of trees from your property. There can be substantial fines for tree removal that goes against regulations.
Stump grinding uses a powerful machine that destroys the stump of the tree and also kills off the root system. A stump grinding machine has a cutting disk that spins at speed which impacts the stump and chops it up.
When a tree is pruned it is usually to try and help the tree and make it healthier and stronger by removing parts of the tree so it can focus its energy on the parts that are going to help it. When tree lopping is performed then it is usually just about removing parts of the tree without much thought as to the negative impacts it may have on the tree.
This will depend on how severe the damage is and where the tree is located. If there is a risk that the tree will impact property or people then it may need to be removed. If however it can be managed and remove the damage parts without causing risk to others then the tree might be able to be cleaned up without having it removed.
The price will depend on how high and large the tree is and where it is located. If it requires extra machinery and is in an inaccessible location then the costs will be higher.