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The answers are the following.

(1) It is well known that the singularity at the vertex of the cone over the Veronese surface is isomorphic to a quotient singularity of type $\frac{1}{2}(1, \, 1, \,1)$, that is the isolated singularity given by the quotient of $\mathbb{C}^3$ by the action of $\mathbb{Z}/2 \mathbb{Z}$ of the form $(x, \,y, \,z) \mapsto (-x, \, -y, \, -z)$. It is no difficult to check the claim directly by looking at the ring of invariants for the action. This is a terminal quotient singularity whose discrepancy is $1/2$ (I think that a detailed computation is carried out in [Reid, Young person guide to canonical singularities]).This means $$K_Y = f^*K_X + \frac{1}{2}E,$$ hence $d=\frac{1}{2}$.

(2) EDIT. This part was corrected by following Sandor Kovacs'observations.

We can identify $X$ with the weighted projective space $\mathbb{P}=\mathbb{P}(1,\, 1,\, 1, \, 2)$. Therefore, by using standard formulas for projective weighted complete intersections (see for instance Dolgachev's paper Weighted projective varieties) we compute the dualizing sheaf of $\mathbb{P}$, obtaining $$\omega_\mathbb{P} = \mathcal{O}_{\mathbb{P}}(-1-1-1-2) = \mathcal{O}_{\mathbb{P}}(-5).$$

Here the (non Cartier) reflexive coherent sheaf $\mathcal{O}_{\mathbb{P}}(1)$ is a generator for the class group $\textrm{Cl}(\mathbb{P})$, whereas the Cartier divisor $H= \mathcal{O}_\mathbb{P}(2)$ is a generator for the Picard group $\textrm{Pic}(\mathbb{P}).$ This shows that $\mathbb{P}$ is $2$-Gorenstein, in fact $$2K_\mathbb{P} = -5H.$$ Finally, the identification of $X$ with $\mathbb{P}$ induces an identification of $H$ with the hyperplane section $\mathcal{O}_X(1)$ of $X$, hence we obtain $$2K_X = \mathcal{O}_X(-5),$$ that is we can take $a=2$ and $b=-5$.

The answers are the following.

(1) It is well known that the singularity at the vertex of the cone over the Veronese surface is isomorphic to a quotient singularity of type $\frac{1}{2}(1, \, 1, \,1)$, that is the isolated singularity given by the quotient of $\mathbb{C}^3$ by the action of $\mathbb{Z}/2 \mathbb{Z}$ of the form $(x, \,y, \,z) \mapsto (-x, \, -y, \, -z)$. It is no difficult to check the claim directly by looking at the ring of invariants for the action. This is a terminal quotient singularity whose discrepancy is $1/2$ (I think that a detailed computation is carried out in [Reid, Young person guide to canonical singularities]).This means $$K_Y = f^*K_X + \frac{1}{2}E,$$ hence $d=\frac{1}{2}$.

(2) EDIT. This part was corrected by following Sandor Kovacs' observations.

We can identify $X$ with the weighted projective space $\mathbb{P}=\mathbb{P}(1,\, 1,\, 1, \, 2)$. Therefore, by using standard formulas for projective weighted complete intersections (see for instance Dolgachev's paper Weighted projective varieties) we compute the dualizing sheaf of $\mathbb{P}$, obtaining $$\omega_\mathbb{P} = \mathcal{O}_{\mathbb{P}}(-1-1-1-2) = \mathcal{O}_{\mathbb{P}}(-5).$$

Here the (non Cartier) reflexive coherent sheaf $\mathcal{O}_{\mathbb{P}}(1)$ is a generator for the class group $\textrm{Cl}(\mathbb{P})$, whereas the Cartier divisor $H= \mathcal{O}_\mathbb{P}(2)$ is a generator for the Picard group $\textrm{Pic}(\mathbb{P}).$ This shows that $\mathbb{P}$ is $2$-Gorenstein, in fact $$2K_\mathbb{P} = -5H.$$ Finally, the identification of $X$ with $\mathbb{P}$ induces an identification of $H$ with the hyperplane section $\mathcal{O}_X(1)$ of $X$, hence we obtain $$2K_X = \mathcal{O}_X(-5),$$ that is we can take $a=2$ and $b=-5$.

The answers are the following.

(1) It is well known that the singularity at the vertex of the cone over the Veronese surface is isomorphic to a quotient singularity of type $\frac{1}{2}(1, \, 1, \,1)$, that is the isolated singularity given by the quotient of $\mathbb{C}^3$ by the action of $\mathbb{Z}/2 \mathbb{Z}$ of the form $(x, \,y, \,z) \mapsto (-x, \, -y, \, -z)$. It is no difficult to check the claim directly by looking at the ring of invariants for the action. This is a terminal quotient singularity whose discrepancy is $1/2$ (I think that a detailed computation is carried out in [Reid, Young person guide to canonical singularities]).This means $$K_Y = f^*K_X + \frac{1}{2}E,$$ hence $d=\frac{1}{2}$.

(2) EDIT. This part was corrected by following Sandor Kovacs'observations.

We can identify $X$ with the weighted projective space $\mathbb{P}(1,\, 1,\, 1, \, 2)$. Therefore, by using standard formulas for projective weighted complete intersections (see for instance Dolgachev's paper Weighted projective varieties) we compute the dualizing sheaf of $X$, obtaining $$\omega_X = \mathcal{O}_{\mathbb{P}}(-1-1-1-2) = \mathcal{O}_{X}(-5).$$ This shows that we can take $a=1$ and $b=-5$.

Here the (non Cartier) reflexive coherent sheaf $\mathcal{O}_X(1)$ is a generator for the class group $\textrm{Cl}(X)$, whereas the Cartier divisor $H= \mathcal{O}_X(2)$ is a generator for the Picard group $\textrm{Pic}(X).$ This shows that $X$ is $2$-Gorenstein, in fact $$2K_X = -5H.$$

The answers are the following.

(1) It is well known that the singularity at the vertex of the cone over the Veronese surface is isomorphic to a quotient singularity of type $\frac{1}{2}(1, \, 1, \,1)$, that is the isolated singularity given by the quotient of $\mathbb{C}^3$ by the action of $\mathbb{Z}/2 \mathbb{Z}$ of the form $(x, \,y, \,z) \mapsto (-x, \, -y, \, -z)$. It is no difficult to check the claim directly by looking at the ring of invariants for the action. This is a terminal quotient singularity whose discrepancy is $1/2$ (I think that a detailed computation is carried out in [Reid, Young person guide to canonical singularities]).This means $$K_Y = f^*K_X + \frac{1}{2}E,$$ hence $d=\frac{1}{2}$.

(2) EDIT. This part was corrected by following Sandor Kovacs'observations.

We can identify $X$ with the weighted projective space $\mathbb{P}=\mathbb{P}(1,\, 1,\, 1, \, 2)$. Therefore, by using standard formulas for projective weighted complete intersections (see for instance Dolgachev's paper Weighted projective varieties) we compute the dualizing sheaf of $\mathbb{P}$, obtaining $$\omega_\mathbb{P} = \mathcal{O}_{\mathbb{P}}(-1-1-1-2) = \mathcal{O}_{\mathbb{P}}(-5).$$

Here the (non Cartier) reflexive coherent sheaf $\mathcal{O}_{\mathbb{P}}(1)$ is a generator for the class group $\textrm{Cl}(\mathbb{P})$, whereas the Cartier divisor $H= \mathcal{O}_\mathbb{P}(2)$ is a generator for the Picard group $\textrm{Pic}(\mathbb{P}).$ This shows that $\mathbb{P}$ is $2$-Gorenstein, in fact $$2K_\mathbb{P} = -5H.$$ Finally, the identification of $X$ with $\mathbb{P}$ induces an identification of $H$ with the hyperplane section $\mathcal{O}_X(1)$ of $X$, hence we obtain $$2K_X = \mathcal{O}_X(-5),$$ that is we can take $a=2$ and $b=-5$.

The answers are the following.

(1) It is well known that the singularity at the vertex of the cone over the Veronese surface is isomorphic to a quotient singularity of type $\frac{1}{2}(1, \, 1, \,1)$, that is the isolated singularity given by the quotient of $\mathbb{C}^3$ by the action of $\mathbb{Z}/2 \mathbb{Z}$ of the form $(x, \,y, \,z) \mapsto (-x, \, -y, \, -z)$. It is no difficult to check the claim directly by looking at the ring of invariants for the action. This is a terminal quotient singularity whose discrepancy is $1/2$ (I think that a detailed computation is carried out in [Reid, Young person guide to canonical singularities]).This means $$K_Y = f^*K_X + \frac{1}{2}E,$$ hence $d=\frac{1}{2}$.

(2) EDIT. This part was corrected by following Sandor Kovacs'observations.

We can identify $X$ with the weighted projective space $\mathbb{P}(1,\, 1,\, 1, \, 2)$. Therefore, by using standard formulas for weighted projective complete intersections (see for instance Dolgachev's paper Weighted projective varieties) we compute the dualizing sheaf of $X$, obtaining $$\omega_X = \mathcal{O}_{\mathbb{P}}(-1-1-1-2) = \mathcal{O}_{X}(-5).$$ This shows that we can take $a=1$ and $b=-5$.

Here the (non Cartier) reflexive coherent sheaf $\mathcal{O}_X(1)$ is a generator for the class group $\textrm{Cl}(X)$, whereas the Cartier divisor $H= \mathcal{O}_X(2)$ is a generator for the Picard group $\textrm{Pic}(X).$ This shows that $X$ is $2$-Gorenstein, in fact $$2K_X = -5H.$$

The answers are the following.

(1) It is well known that the singularity at the vertex of the cone over the Veronese surface is isomorphic to a quotient singularity of type $\frac{1}{2}(1, \, 1, \,1)$, that is the isolated singularity given by the quotient of $\mathbb{C}^3$ by the action of $\mathbb{Z}/2 \mathbb{Z}$ of the form $(x, \,y, \,z) \mapsto (-x, \, -y, \, -z)$. It is no difficult to check the claim directly by looking at the ring of invariants for the action. This is a terminal quotient singularity whose discrepancy is $1/2$ (I think that a detailed computation is carried out in [Reid, Young person guide to canonical singularities]).This means $$K_Y = f^*K_X + \frac{1}{2}E,$$ hence $d=\frac{1}{2}$.

(2) EDIT. This part was corrected by following Sandor Kovacs'observations.

We can identify $X$ with the weighted projective space $\mathbb{P}(1,\, 1,\, 1, \, 2)$. Therefore, by using standard formulas for projective weighted complete intersections (see for instance Dolgachev's paper Weighted projective varieties) we compute the dualizing sheaf of $X$, obtaining $$\omega_X = \mathcal{O}_{\mathbb{P}}(-1-1-1-2) = \mathcal{O}_{X}(-5).$$ This shows that we can take $a=1$ and $b=-5$.

Here the (non Cartier) reflexive coherent sheaf $\mathcal{O}_X(1)$ is a generator for the class group $\textrm{Cl}(X)$, whereas the Cartier divisor $H= \mathcal{O}_X(2)$ is a generator for the Picard group $\textrm{Pic}(X).$ This shows that $X$ is $2$-Gorenstein, in fact $$2K_X = -5H.$$