Blow-up $\mathbb C^2$ simultaneously at all points of $\mathbb Z \times \{0\}$.
Part a) is evident for the blown-up manifold $X$ since it contains one-dimensional compact submanifolds. As for part b), I am guessing (but cannot prove) that the huge first Chern class of the line bundle $\mathcal O_X(D)$ associated to the exceptional divisor $D$ prevents the existence of a finite number of holomorphic or even differentiable charts.