Joel showed you how to get infinitely many non-$T_0$ non-homeomorphic spaces as you want. But you can even get infinitely many $T_0$ non-homeomorphic spaces as you want just by isolating points $0$ through $k$ in your topology. The resulting space will still have the property you want and exactly $k$ many isolated points.

You cannot wish to get a Hausdorff, not even a $T_1$ space as you want because a space is $T_1$ if and only if singletons are closed. So, a finite set in a $T_1$ space will only generate finitely many sets under the operations of closure, complement and union.

Joel showed you how to get infinitely many non-$T_0$ non-homeomorphic spaces as you want. But you can even get infinitely many $T_0$ non-homeomorphic spaces as you want just by isolating points $0$ through $k$ in your topology ($k \geq 1$). The resulting space will have exactly $k$ many isolated points and every $\{j\}$, for $j \geq k$ can be generated from the finite set $\{0,1, \dots, k-1\}$ by closure, complement and union.

You cannot wish to get a Hausdorff, not even a $T_1$ space as you want because a space is $T_1$ if and only if singletons are closed. So, a finite set in a $T_1$ space will only generate finitely many sets under the operations of closure, complement and union.