This question requires no calculations.
You should not integrate anything to answer it.
$\phi(x)$ is the density function of a distribution $D$,
$\Phi(x)$ is the cumulative distribution function of $D$,
the density $f(x) = 2\phi(x)\Phi(x)$
corresponds to the distribution of a random variable $X$ defined
as the greater of two random variables (say $D_1,D_2$) picked according to $D$.
Why you would see this
To see this from the formula for $f$, see that $\phi(x)$ corresponds to the chance that $D_1$ has a certain value, and $\Phi(x)$ is the chance that another sample $D_2$ has a lesser value. Finally, the factor of 2 is for the alternative case that actually $D_2$ had this value and was greater, so together, $f(x)$ is the chance that $x$ is the greater of two randomly picked values.
Once you see this, answering the question is easy and requires no calculations.
A: Yes, you expect the higher of two samples to be greater on average than what you expect for just the first sample, since the second sample can only make the maximum go up. (Remember $0$ is the expected value of a single sample of $N(0,1)$.)
B: No, you don't expect it to be lower.
C: Will the greater of two samples of $N(0,1)$ be negative more than half the time? Of course not! (It will be negative 1/4 of the time -- when both samples are negative.)
D: Will the greater sample be positive less than 1/4 of the time? Of course not! (It will be positive 3/4 of the time -- when either sample is positive.)