Yes to your first question, “not quite” to the second. Laws of gravity are the same for big and small object. However, these laws state that the force between objects is proportional to their mass and inversely proportional to their distance squared (distance * distance).
So, they can still orbit each other (any pair of objects in space can orbit each other if their relative speed allows it) but just dividing all sizes in half (unless you mean just compressing the Earth and the Moon while keeping the mass and distance intact) will screw up the equations, so their orbital speeds and parameters of the orbit will be different.
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