You don't tell me if this floor is a concrete slab or a frame construction, but I am guessing it is frame. If it is concrete get a good grinding cup and grind it to level. Here is a fast working cup wheel:
http://www.defusco.com/5811-brazed-concave-cup-p-2008.htmlYou also mention quartz countertops, and since Home Depot doesn't sell these I am thinking you ordered a counter made from engineered stone, which is 93% quartz and 7% resin. These work just like granite and have a lot of advantages over granite, but that is a personal preference as I prefer natural stone. That being said, and assuming that you have indeed purchased counters made from engineered stone, let's look at this systematically:
First, if you can level the floor that is the obvious choice, as it will have to be levelled at some point to put a tile floor down - be it ceramic tiles, porcelain tiles or stone tiles (engineered or natural). Once the floor is close to level a good stone setter should be able to make the floor work for you. If the people from Home Depot can't do it, perhaps you should check locally for someone who specializes in stone floors. Don't be afraid to ask for references, and check them once you have them.
OK, now on to your questions:
1. What would people recommend doing to reduce the potential damage the cabinets and countertops?
There are no shortcuts here - if you build with a bad foundation your roof won't stay up, and if the flooring and/or cabinets are not stable your counters may shift and crack.
2. Could the floor sinking under that cabinets crack my countertop or make the countertop break at the seams?
Yes, or worse - maybe the stress causes a break in the center of the slab.
3. If a crack occurs, can the countertops (which have been glued to the walls and cabinets with silicone) be removed without extensive damage to the cabinetry?
Is there a substrate between the cabinets and the counters? If they are siliconed to the exposed cabinet ribs and you can access them you can saw through the silicone with a razor blade. I have done this and it is extremely difficult at best. If you don't have good access or if there is a substrate like plywood or cement board that is it attached to it can be impossible. You may even break them if you totally destroy the cabinets.
4. How do you remove stone countertops?
It depends on how they are attached and what they are attached to. Feel free to get specific on this and send me a PM or just call me at 800-289-6834.
Sorry I have no good news for you yet, but please call me and I will help you troubleshoot this situation.
Glad to be of service,
Mark