As to cutting or pushing dead trees. Dead trees have dead branches that like to fall down on you and/or your tractor when you least expect it!
I haven't cut down a tree in the last 2 years that wasn't at least 80 feet tall, and I can tell you that I don't dig around them and try to push them over.
I tie a good 1" nylon rope up as high as I can get it, run that out to a snatch block/pulley, and then the end out to an area that I can hook it up to a truck. My helper then pulls the rope until it starts to pull the truck back, brakes the truck and waits while I cut the tree.
As soon as I'm close to the final cut, I signal the helper and he starts accelerating the truck. Once the tree moves the least bit toward the fall, it's gone, but he continues pulling in the beginning to make sure of no surprises.
A combination of the wedge cut and the back cut "should" cause the tree to fall where I want it, but anything can happen, so the rope is essential.
If it's convenient to, it's best to leave several feet of trunk, especially on pines, so that you can dig and push out the rest of it later, but keep in mind that cutting up high may make it more difficult to get out of the way when the tree begins to fall.
That is a critical time because if you haven't made the right cuts, and if you cut all the way through instead of leaving a small piece intact, the tree can kick back and up on you. Cutting high means you're probably on a ladder, and you may have a hard time getting out of there in time.
Rather than continue with this, here's a very good, short and simple explanation of cutting a tree:
Cutting a tree
John