Can the topology $\tau = \{\emptyset, \{a\}, \{a,b\}\}$ be induced by some metric?

Every metric space is Hausdorff. This topology is not. Every neighbourhood of $b$ contains $a$. So $\tau$ is not induced by any metric.

It is easy to prove that every topology $\tau$ induced by a metric space $(X,\delta)$ on a finite set $X$ is discrete.

How? Just prove every point is open, to do this notice that $\{x_0\}=B(x_0,d)$. Where we define $d$ as $\min\limits_{x\in X-\{x_0\}} \delta(x,x_0)$