Unclear time-to-frequency integration step From here; $\hat f=\mathcal{F}(f)$, bar = complex conjugate:


Time-shift property: $x(t-b) \Leftrightarrow e^{-j\omega b}{\bf X} (\omega)$, so why is it $+$ (red)?
What at all is happening? Looks like convolution theorem, taking $f$ and $\psi$'s time-domain multiplication into frequency domain, except don't we need a second integral for conv.?
That's an application of Parseval's (Plancherel's) Theorem:
$$\int_{-\infty}^{\infty}f(t)g^*(t)dt=\frac{1}{2\pi}\int_{-\infty}^{\infty}F(\omega)G^*(\omega)d\omega\tag{1}$$
where $F(\omega)$ and $G(\omega)$ are the Fourier transforms of $f(t)$ and $g(t)$, respectively. I've used $^*$ to denote complex conjugation.