Questions about answer given for Calderón reproduction formula proof

Question

I have some questions about this question and in particular the fist answer given by user77935.

1. In the 7th equation, he makes two estimates of the form $$ \int_0^{\xi^{-1}} \widehat \phi(t\xi)^{-2} \frac{\text{d}t}{t} \leq ||\nabla \widehat \phi||_{L^\infty}^2 \int_0^{\xi^{-1}} (t\xi)^{-2} \frac{\text{d}t}{t}. $$ Where does this $||\nabla \widehat \phi||_{L^\infty}^2$ come from, why does this estimate hold?
2. After that, he states that $$ |\widehat f_R(\xi) - \widehat f(\xi)|^2 \leq |\widehat f(\xi)|^2 ( 1 + ....) $$ is an $L^1$ function. What exactly is the $L^1$ function here and why is it important?
3. The last step is applying dominated convergence. However, to me it appears that for $\lim_{R\to \infty} || \widehat f_R - \widehat f||_{L^2}^2 = 0$ to hold I have to assume that $$ \int_0^\infty \widehat \phi (t\xi)^2 \frac{\text{d}t}{t} =1 $$ at some point?

Thanks for any help on clarifying these questions!