Let $X$ be the value of the first dice and $Y$ the sum of the values when two dices are rolled. What is the correlation of $X$ and $Y$?

Question

Let $X$ be the value of the first dice and $Y$ the sum of the values when two dices are rolled.

1. What is the correlation of $X$ and $Y$?
2. Do $X$ and $Y$ have functional dependency?

What I have gotten so far:

1. $$Corr(X,Y)=\frac{Cov(X,Y)}{\sqrt{Var(X)\cdot Var(Y)}}$$

Since

$$P(X=x,Y=y)=\frac{1}{36}\neq\frac{1}{216}=\frac{1}{36}\cdot\frac{1}{6}=P(X=x)P(Y=y),$$

$X$ and $Y$ are not independent hence $Cov(X,Y)\neq 0$.

What I need to solve now are

$$Cov(X,Y)=E(XY)-EXEY$$ $$VarX=EX^2+(EX)^2$$ $$VarY=EY^2+(EY)^2.$$

However, I don't know how to start solving $E(XY)$ in this case. We haven't learned about moment-generating functions yet.

2. $X$ and $Y$ are functionally dependent if we know the value of $X$, then we also know the value of $Y$. I don't really know how to start proving the randomness of $Y$ even if we knew $X$. Is there any ways of proving this with contradiction?