If your contractor also installed a new threshold and frame with the new front door then air and water inflitration should not be a problem if the job was done properly. Ideally when installing a new front door you would replace both the door and the frame at the same time. This is because the door, frame and threshold are matched at the mill. Alternatively you might have a door custom morticed and trimmed to fit an existing frame and threshold. Since your contractor cut into the floor, I assume that he also installed a new threshold and frame. That being the case he probably had to cut the floor to accomodate either a taller door frame than the original or a deeper threshold. In either case he should have sealed below the threshold with polyurethane caulk (NP1 is what I use) and around the frame with polyurethane foam. If this was done then air and water will not come into the house. If this was not done then you will need to pull the casing (trim) from the door frame, foam the space behind the casing between the wall and the door frame, reinstall the trim and then caulk the threshold to the floor (its not worth pulling the threshold to seal below it at this point). As for the trench that he created in your floor, you are right on that you could use LevelQuik or any other self leveling cement. However these cements are very expensive when compared to other options and you really only need a few handfuls for your purposes. My suggestion would be to fill the trench with the cheapest thinset you can find. CustomBlend (Home Depot) is $4-5 a bag and will work great. You will have to trowel it smooth whereas the LevelQuik levels itself to a large degree but you will save yourself about $20 and therefore you won't feel so bad when you only use a couple of handfuls out of a 50# bag. The baby thrshold would also work and if you would want a more aesthetically pleasing finish you could add the baby threshold on top of the thinset once it has cured. From what you said about having the floor replaced in the future, I would not use the baby threshold as it will just be thrown away. I would also not put more thinset into the trench than is required to fill to the level of the concrete under the asbestos as you will just have to chip it out once you remove the old floor and install a new one. When you decide to replace the floor I would suggest not just covering the old one since you are already at least 1/4" up on your threshold. If you continue adding new floors on top of the old one, the sweep on the bottom of the door may drag across your floor and you most likely will be unable to place a door mat on the inside of your door. I would suggest that you insist that the old floor be removed. Good luck.
General C&ontractor
Extensive experience in carpentry, plumbing, electrical and all other aspects of construction.