Yes they should do. Maybe not 2x with every cubid, but definitely exponentially - for larger "state space" as you call it. This is what makes them different from classical model.
Not generally. Only for certain problems. There are precious few problems that quantum computers are known to be asymptotically better at than classical computers, and one of them (Grover's search) is only a sqrt(N) speedup.
Just speaking in terms of "quantum size" (and not computing power, which is not understood yet), it depends on the graph of possible entanglements among the qubits. If it were a complete graph (allowing arbitrary entanglement) then the size of the relevant Hilbert space would exactly double with each additional qubit (i.e. one would need twice as many complex numbers to describe any particular wave function in the space). But the D-Wave chips operate with a fixed topology that (I think) is far from fully-connected (e.g. the 128-bit chip used a "Chimera graph", which they describe in blog posts and publications). The growth would only be truly exponential with sufficient connectivity (e.g. a planar graph would mean sub-exponential growth).
