Blow-up $\mathbb C^2$ simultaneously at all points of $\mathbb Z \times \{0\}$. Part a) is evident for the blown-up manifold since it contains one-dimensional compact submanifolds. As for part b), I think that the huge first Chern class of the normal bundle to the exceptional divisor prevents the existence of a finite number of holomorphic or even differentiable charts.
Georges Elencwajg
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