Is an allergy to dust mites the same as an allergy to dust I can see on the floor?

Yes. Dust mites are usually components of the dust that you can see in your home.
Not necessarily. Dust mite droppings may be part of the dust on the floor, but floor dust may also contain mold, mildew, pollen, animal dander. To learn to what you are specifically allergic, see an allergist for testing. If allergies are severe, doc may recommend desensitization through series of diluted allergens.
No. Check out this link.