How Do You Treat A Cut Tree Limb So The Rest Of The Limb Will Not Be Damaged?

I'm assuming you're intending to not damage the main tree when the branch falls. First you make a partial cut on the UNDER side of the branch. This will act as a stop-cut that doesn't allow the falling branch from ripping bark and wood from the main tree. You must be careful not to make too deep of a preliminary under-cut which can bind your saw if the branch sags too much.

Paint the exposed part to seal it. You can use polyurethane or just a can of spray paint.

If I cut a branch off a hedge do I hae to apply some kind of chemical to help the hedge heal?