A discrete topological space is a space where all singletons are open $\implies$ all sets are clopen? Closed?

Let $F$ be an arbitrary subset of $X$ where $X$ is equipped with discrete topology.

As you said: in a discrete topological space all singletons are open.

As you said: arbitrary unions of singletons are open so $F^c=\bigcup_{x\in F^c}\{x\}$ is an open set.

(You don't even need this subroute: in a discrete space all sets are open by definition)

Then its complement $F$ is a closed set.