Approach the covariance matrix I have this problem:

Let the random vector $\binom{X}{Y}\sim
N_{2}(\binom{0}{0},\bigl(\begin{smallmatrix} 1 & \rho\\  \rho & 1 \end{smallmatrix}\bigr))$
1) Find the distribution of $Z=X+Y$.
2) Find the distribution of $W=X^2$, the mean and the variance.
3) Calculate $Cov(X,W)$ and $Cov(Z,W)$.

It's the first time that I have found myself analysing this type of function. What the covariance matrix involves? How do I manage it? Thanks in advance for any clarification!

I will not post full answers, so that You can practice a bit - as it is Your first encounter of this subject. Nevertheless I will point out some crucial facts, that are needed here if one need to handle this.

$X=(X_{1},\cdots, X_{k})^{T}$ is multivariate normal distributed if and only if every linear combination of the form $$Y=a_{1}X_{1}+\cdots+a_{k}X_{k}$$ is normally distibuted (well known, one dimensional distribution).

From that we see, that $Z=X+Y$ must be normally distributed. Recall, that if $Z$ is normally distributed, then its distribution is uniquely determined by its mean $\mu$ and variance $\sigma^{2}$. Variance is basically "Covariance matrix" - which in this case is one dimensional. And here comes the second fact.

If $C$ is covariance matrix of $X=(X_{1},\cdots, X_{k})^{T}$ and $a\in \mathbb{R}^{k}$, then $$\mathrm{Cov}(a^{T}X)=a^{T}Ca.$$

Knowing this You can proceed with point a).
In b) You are asked about distribution of $W=X^{2}$, which depends only on distribution of $X\sim \mathrm{N}(0,1)$. This is a very famous distribution. This problem does not deal with any multivarious stuff. 
Point c) $\mathrm{Cov}(X,W)=\mathrm{C}(X,X^{2})$. You can find it by basic calculations. From linearity of Covariance we have
$$\mathrm{Cov}(Z,W)=\mathrm{Cov}(X+Y,X^{2})=\mathrm{Cov}(X,X^{2})+\mathrm{Cov}(Y,X^{2}).$$
Only the last part $\ \mathrm{Cov}(Y,X^{2})=\mathbb{E}(YX^{2}), \ $
is a little bit tricky. 
The idea is that we can (without loss of generality) express $Y$ as $$Y=c_{1}X+c_{2}V$$  where $V\sim \mathrm{N}(0,1),$ $$c_{1}^{2}+c_{2}^{2}=1$$ and $X,V$ are independent.
Ask if You will need more explanations.
