Hi Redsun,
I fully agree with the above posts but would also like to add a couple of tips that you may find helpful.
I think your right about the roots lifting the slabs. If you do decide to take the root out as well, you may find it easier if you leave a tall stump about 3ft high, this will give you some extra leaverage (waggle it about like a loose tooth)
I dont know how you are at felling trees but obviously you dont want it falling towards the house, to avoid this cut a wedge out of the side where you want it to fall first and then start to saw from the back, you can assist this by tying a rope about 3/4 of the way up with someone keeping it tought at a safe distance, if its only 8-9ft you should find it easy enough but the root will be the hardest part and if the groud is bone dry,like most at the moment,you could drench the ground before you start to dig to soften it up.
Take care T bird.