are you saying that the floor has started to crack after the tree was cut down?
This is more characteristic of ground heave, common on clay soils. What happens is that the old tree used to suck moisture out of the ground, making it shrink. When the tree is cut down, the ground gradually absorbs water again and expands back to its "normal" size. If the garage was built when the tree was big, it will have been built on shrunken, subsided soil, and it is going to get pushed up when the soil expands again.
I am going to guess that the garage was built on a simple unreinforced concrete raft, possibly with thickened edges under the walls if brick. Unlike a house foundation, it will not have been dug deep enough to avoid heave, it is just lying on the surface, or maybe six to ten inches deep.
If my guesses are correct, then I don't see that you can do anything to stop the ground heaving and lifting the garage floor. You might consult a builder about having the concrete raft broken up (except under the walls) and some paving slabs laid on sand on a dpm, at least this will not suffer any serious damage from heave, though it may bulge a bit and become uneven. In which case the slabs can be lifted, and relaid after levelling the sand.
It will take several years for the ground to finish moving, no point in spending a lot of money repairing if it's going to move again.
As regards the stump not being dead, you can carve a bowl into it and fill it with SBK brush killer or other chemical, which will soak in and kill it (protect from rain). You can have it chipped if you want, but still need to kill any regrowth or suckers by cutting them off quickly and dabbing SBK or Glyphosate on the stumps. The roots will NOT grow bigger now the tree has been cut down, they will gradually rot away (this may take 20 years or more)