I just got new granite counter tops in my kitchen. I am not sure about the cutting - but to be safe use a cutting board.
Granite is a natural stone surface.
You cannot put hot pans directly on the surface or leave wet dishes on the surface for lengthy periods of time - stone is a porous surface so immense heat will scorch the surface and water will leave a water mark. Room by Room hosts advised that you should have it professionally resealed every year and wax it on a monthly basis. Don't be dismayed. Remember granite will last forever if you treat it right and it will look fantastic to boot. I am sure you can find a website on care for granite countertops or call the granite manufacturers themselves.