I do not know how much help I could be but I think automatic may be referring to a CVT style drive that has two pulleys and a belt (like a snowmobile) and the hydrostatic is a fluid drive. There are likely pros and cons to each which I could not tell you. It would take some research into the different drive manufacturers to really tell them apart; meaning drive speed, axle torque and so on. As for Sears vs Deere, since Sears do not make their own mowers, I have never really liked them, I figure I would just buy directly from MTD or Husqvarna, but when they are discounted $300-500 that is quite significant. The Deere, although a D series, should be more than enough for you 1/2 acre. Just take care of it and keep up on oil, lube, wearables and it should last. I am not a Deere fan at all, but if these were my only three options, that is what I would go with. Hope this helps a bit.
Mike