Prove a random variable has normal distribution 
Let $X$ be a standard normal random variable and $a>0$ a constant.
  Define:
$Y = \begin{cases}
      \phantom{-}X & \text{if $\,|X| < a$}; \\
     -X & \text{otherwise}.
\end{cases}$
Show that $Y$ is a standard normal distribution and the vector $(X,Y)$
  is not two-dimensional normal distributed.

I tried approaching this using the expectation saying for $g$ a measurable function
\begin{align*}
E[g(Y)] 
  &= \int_{\mathbb{R}} g(y)\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}} dx  \\
  &= \int_{-\infty}^{-a} g(x)\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}} dx + \int_{-a}^a g(-x)\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}} dx + \int_a^{\infty} g(x) \frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}} dx
\end{align*}
as $x^2 = (-x)^2$ I could reason that $Y$ must have normal distribution, but this just doesn't seem clean enough.
As to the second question I thought I could express $(X,Y)$ as a function of $X$ and then:
$E[g(X,Y)] = E[g(h(X))] = E[f(X)]$ where $f = g \circ h$ a measurable function at which point I was confused by the dimensions.

The distribution of a real-valued random variable $Y$ is determined by its cdf $F := \mathbb{P}(Y \leq y)$ (because sets of the form $\{(-\infty, y]\}$ generate the Borel $\sigma$-algebra on $\mathbb{R}$). Let $N$ denote a standard normal random variable, $Y = -X$ and $y \in \mathbb{R}$; there are three cases: (i) $y \leq -a$, (ii) $-a < y \leq a$, and (iii) $a < y$. Consider case (ii):
\begin{align*}
P(Y \leq y) &= P(Y \leq -a) + P(-a < Y \leq y) \\
&= P(-X \leq -a) + P(-a < X \leq y) \\
&= P(X \geq a) + P(-a < N \leq y) \\
&= P(N \geq a) + P(-a < N \leq y) \\
&= P(N \leq -a) + P(-a < N \leq y) \\
&= P(N \leq y).
\end{align*}
The first equality follows by disjointness and finite additivity; the second because $\{\omega : Y \leq -a\} = \{ \omega : -X  \leq -a\}$ and $\{-a < Y \leq y\} = \{-a < X \leq y\}$ for $-a < y \leq a$. The second-to-last line follows by symmetry of $N$. 
The case (i) is easy and (iii) is symmetric. 
For the second question, have you drawn the support of the measure on $\mathbb{R}^2$? 
