Are there any countable Hausdorff connected spaces?

Do countable Hausdorff connected topological spaces exist?

Yes, see here (MathOverflow) for references to some non-trivial examples.

Trivially yes, a singleton for example ;). Non-trivial examples abound, for example ''A countable connected Hausdorff space'' by Brown, in Bull. Amer. Math. Soc., 59 (1953) p. 367.

$\pi$-Base is an online encyclopedia of topological spaces inspired by Steen and Seebach's Counterexamples in Topology. It lists the following countable, connected, Hausdorff spaces. You can learn more about any of them by visiting the search result.

Gustin's Sequence Space

Irrational Slope Topology

Prime Integer Topology

Relatively Prime Integer Topology

Roy's Lattice Space