Question 1 is interesting but, I'll guess, very difficult?

The answer to question 2 is "no".  This is because *any* finitely presented group appears as the fundamental group of *some* compact four-manifold (without boundary).  I suppose that I also need to display an infinite collection of quasi-isometry types of finitely presented groups so... $\mathbb{Z}^n$.

The answer to question 3 is "no".  Consider $S^2 \times S_2$, the two-sphere crossed with the surface of genus two.  

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Perhaps you want to add the hypothesis that the four-manifold is aspherical: that is, the four-manifold is a $K(G, 1)$.  I suspect that the answers to 2 and 3 will remain "no"?