Note that you have the pdf of the standard normal not quite right, or at least not consistently right. The constant in front should be $\frac{1}{\sqrt{2\pi}}$. So the constant in front of the joint density is $\frac{1}{2\pi}$.
Let random variable $R$ denote the distance to the origin. Then $R=\sqrt{X^2+Y^2}$. We want the cumulative distribution function (cdf) of $R$, so we want $P(R\le r)$. Clearly this is $0$ for $r \lt 0$. So from now on assume $r\ge 0$.
We have
$$P(R \le r)=\iint_{D_r} \frac{1}{2\pi}e^{-(x^2+y^2)/2}dx\,dy,$$
where $D_r$ is the disk of radius $r$, centre the origin. We need to evaluate this integral.
To do this, change to polar coordinates. (If that has been forgotten, look back to your several variable calculus course.) Because the letter $r$ is already taken, I will use $t$ for the polar coordinate distance to the origin. Let $x=t\cos\theta$, $y=t\sin\theta$. The integral becomes
$$\int_0^{2\pi} \int_0^r \frac{1}{2\pi}e^{-t^2/2}t\,dt\, d\theta.$$
The integration is straightforward. The inner integral is $\frac{1}{2\pi}\left(1-e^{-r^2/2}\right)$, and for the outer integral you just multiply by $2\pi$, getting $1-e^{-r^2/2}$.
Now we have the cumulative distribution function of $R$. The rest follows without much trouble. For (c), differentiate. For (d), if the cdf is $F_R(r)$, we want $F_R(3)-F_R(2)$.
That is right. There is a more general way to calculate such affine transformations you probably will see later. We have already established that
$$ \begin{pmatrix} X\\ Y \end{pmatrix} \sim N_2\left(
\begin{pmatrix} 0 \\ 0 \end{pmatrix} ,
\begin{pmatrix} 1& \rho\\ \rho&1 \end{pmatrix}\right) $$
then $aX+bY+c = (a,b)(X,Y)^T + c$ and we find
$$ aX+bY+c \sim N\left( c + (a,b) \begin{pmatrix} 0 \\ 0 \end{pmatrix}, (a,b) \begin{pmatrix} 1& \rho\\ \rho&1 \end{pmatrix}\begin{pmatrix} a \\ b \end{pmatrix} \right) = N(c, a^2+b^2 +2ab\rho). $$
This is from a general calculation of affine transformation $Y = \eta + BX$, where $X\sim N_p(\mu,\Sigma)$ is p-dimensional normal distributed, $B$ is a $k\times p$ matrix and $\eta\in \mathbb{R}^k$. Then
$$Y \sim N_k(\eta+B\mu, B\, \Sigma \,B^T). $$
Best Answer
What you're looking for is the Rayleigh distribution (distribution of the norm of two centered and independent gaussian RVs) : http://en.wikipedia.org/wiki/Rayleigh_distribution
You might also want to look up the $\chi^2$ distribution (distribution of the squared norm) : http://en.wikipedia.org/wiki/Chi-squared_distribution