How many "co-" topologies are out there? So far I have learned about $\tau_{co-finite}$ and $\tau_{co-countable}$ Are there any other co-related topologies like...$\tau_{co-infinite}$? In general, what is the condition we need to have a co-topology?

If $\mathcal{C}$ is any collection of subsets of a set $X$, you could try to define a "co-$\mathcal{C}$" topology on $X$ whose open sets are exactly the complements of elements of $\mathcal{C}$. This actually is a topology iff $\mathcal{C}$ satisfies the axioms for closed sets (which are just "complemented" versions of the axioms for open sets): $\mathcal{C}$ must be closed under arbitrary intersections and finite unions.