What is an example of a smooth surface $X$ with an irreducible curve $C \subseteq X$ with $C^2<0$ but $C$ is not rational? I don't know how to make curves with negative self intersection other than by blowing-up, but that gives a rational curve.
I think if follows that $K_X \cdot C>0$ so $X$ can't be Fano.
This question occurred to me in light of the lemma that an embedded curve $C \subset X$ is rational if and only if $C^2<0$ and $K_X\cdot C<0$.
On any surface $X$, take any non-rational curve $C$. Blow up $n$ smooth points on $C$. Then the proper transform of $C$ on the blowup is isomorphic to $C$, but has self-intersection $C^2-n$. So if $n \geq C^2+1$, this gives what you want.
You're right that any such curve must have $K_X \cdot C >0$.
The last sentence is not true, though: there are plenty of rational curves in surfaces with $C^2>0$. Take for example a line in $\mathbf P^2$.