There are (at least) two different types of stains for concrete. One is a surface coating, the other is acid etching.
The former can work, but isn't as durable/lasting. I believe most of the products you get at the big-box stores are more of a top-coat type of stain.
Acid etching is just that, it's a stain containing an acid that actually etches the concrete. This is the typical type of stain you see in home magazines and such. The etching provides a deeper stain that is more durable and not prone to flaking.
I've done the latter, and applying it is super-easy. It's the prep work that can be a chore--namely if you have adhesive on the concrete already. In our case, we has tile mastic, which we had to dissolve using soybean based stripper. It worked great but was incredibly messy.
If you don't have glue to deal with, however, you should be good to go. Order up the stain (lots of places online to get it). You then dilute it per instructions and spray it on the concrete. I used a garden sprayer for this. You then let it sit for a period of time, and then you apply a neutralizer to stop the reaction.
After that, you apply water and mop/squeegee/wet-vac it up.
Then, you apply a sealer of some sorts. Common options are wax-based, polyurethane, and epoxy. I went with poly as it was relatively easy to apply (roll it on). That said, I don't think it was ever truly waterproof. For your needs, you may want to go the epoxy route.
The concerns I'd have -- outside of whether there's anything under the floor that could be disturbed, and whether there's a rock layer under the floor that would prevent your lowering it -- would be:
You're going to have to make sure this doesn't disturb the foundation. I'd suggest getting an engineer's advice before doing anything.
You're going to have to dispose of all that rubble. That means renting a dumpster, and hauling it out of the basement. Not impossible -- a friend of a friend put a basement under an old farmhouse by digging outward from the root cellar, putting the house up on jacks as he went, and then getting a foundation installed under the house, and he did almost all of the digging and hauling himself -- but it's a nontrivial amount of work. You're going to be moving several tons at least.
Putting in the new floor is going to require establishing a good seal against water infiltration between it and the foundation walls. Again, I'd suggest getting an engineer's advice.
So: Yes, theoretically it can be done. Personally, given the cost and effort and potential liability involved I wouldn't consider doing it on property I didn't own. And I definitely wouldn't consider it without expert advice.
Best Answer
With the epoxies I have used minor imperfections like small divots from a heavy object chipping the floor are usually filled in. The epoxy can make a very slick surface. For instance, I coated one bay of my shop that I do automotive work in and wanted it like glass so oil spills were easier to clean. This worked great until I spilled some antifreeze. The antifreeze was so slick I could not stand on it until rinsed several times. I ended going back and added another coat with some traction powder. It is still smooth and easy to clean but not as slippery.
The epoxy did fill minor divots where heavy items had chipped the surface.
As a note of caution make sure to get epoxy specifically made for garage flooring. I always use 2 part and have excellent long term results. A friend just used a basic epoxy paint not specifically made for garages and when he brought his truck in and parked it the heat from the tires caused the epoxy to fail under the tires he had spots all around where he parked that the epoxy peeled off. He ended up having the floor shot blasted to remove the epoxy and recoated with a 2 part garage epoxy then had no further problems.