Show that $f_{\alpha}(t)$ is a p.d.f.

A probabilistic interpretation: consider $(X,Y)$ i.i.d. standard normal, then $\Phi(at)=P(X<at)$ and $\varphi$ is the PDF of $Y$ hence $$\int_{-\infty}^\infty 2\varphi(t)\Phi(at)dt=\int_{-\infty}^\infty 2\varphi(t)P(X<at)dt=2P(X<aY)=2P(Z<0)$$ where $Z=X-aY$ is centered normal with nonzero variance. Every centered normal distribution is symmetric hence $P(Z<0)=\frac12$. This shows that the integral on the LHS is $1$, as desired.