Let $X$ be a rationally connected smooth projective variety defined over $\mathbb C$.
(1) Can we find a surface $S \subset X$ such that $ (-K_X)^2 \cdot S > 0 ? $ If yes, can we assume that $S$ intersects properly a given codimension $2$ subvariety $V \subset X$ only at isolated points ? The answer is obviously no, as Artie pointed out in the comments.
(2) Can the square of the first Chern Class of $K_X$ be numerically equivalent to $\sum \lambda_i Y_i$ where $\lambda_i \in \mathbb Q_{<0}$ are negative rational numbers, and $Y_i$ are irreducible codimension two cycles ?
Edit : As Artie and Francesco noted, (1) is too much to ask for. I still would like to know if (2) can hold ?
Edit 2 : The answer to (2) is yes. If we blow up a point in Francesco's example then we obtain a $3$-fold $Y$ with $K_Y = -F + 2E$. Thus $K_Y^2$ is numerically equivalent to $-4 \ell$, where $\ell$ is a line inside the exceptional divisor $E$.