The answer to the first question is the following: if $X$ has klt singularities, then there exists a $\mathbb{Q}$-factorial variety $Y$ with a birational morphism $\pi: Y\to X$, such that if you write $\pi^*K_X=K_Y+\Delta$, then $\Delta$ is effective. And for any exceptional divisor $E$ of $Y$, we have the discrepancy $a(E,Y,\Delta)>0$, i.e., $(Y,\Delta)$ is terminal. This implies $Y$ itself is terminal, too. If you start with $X$ with only canonical singularities (which is stronger than klt singularities), then $\Delta=0$. This follows from Corollary 1.4.3 of [BCHM]. In fact, it was known before that certain part of MMP would imply the existence of terminalization. The cases of MMP established in [BCHM] contains this part.

The answer to the first question: if $X$ has klt singularities, then there exists a $\mathbb{Q}$-factorial variety $Y$ with a birational morphism $\pi: Y\to X$, such that if you write $\pi^*K_X=K_Y+\Delta$, then $\Delta$ is effective, and for any exceptional divisor $E$ of $Y$, we have the discrepancy $a(E,Y,\Delta)>0$, i.e., $(Y,\Delta)$ is terminal. This implies $Y$ itself is terminal. If you start with $X$ with only canonical singularities (This is stronger than klt singularities. But if $K_X$ is Cartier and $X$ is klt, then $X$ has canonical singularities. I mention this since I know in some cases from the representaion theory, indeed $K_X$ is trivial.), then $\Delta=0$. This follows from Corollary 1.4.3 of [BCHM]. In fact, it was known before that certain part of MMP would imply the existence of terminalization. The cases of MMP established in [BCHM] contain this part.