The following is the proof taken from Lemma $13$.
Questions:
$(1)$: What is the Lipschitz-norm of $\phi_a$? The following is my attempt:
$\| \phi_a \|_{Lip} = \sup_{x \neq y}{\frac{|f(a) - \bar{g}(a)| |\tau(\| x - a \| - \tau(\| y - a \| |}{\| x - y \|}} = |f(a) - \bar{g}(a)| \sup_{x \neq y}{\frac{|\tau(\| x - a \| - \tau(\| y - a \| |}{\| x - y \|}}$. My guess is $\| \phi_a \|_{Lip} = |f(a) - \bar{g}(a)| \| \phi_a \|_{Lip}$, but I couldn't show it.
$(2)$: How to show a function $f$ is $K$-Lipschitz? Is it enough to show that $|f(x)-f(y)| \leq K |x-y|$ for some constant $K$? Assume that $f$ is a real-valued function defined on $\mathbb{R}$.
$(3)$: Why can we assume that $K>1$?
(1) See the Wikipedia entry on Holder spaces. The Lipschitz norm is the special case $\alpha = 1$.
(2) Yes, this is what it means to prove that $f$ is $K$-Lipschitz.
(3) If $f$ is $M$-Lipschitz, then $f$ is also $K$-Lipschitz for any $K>M$. So if you know $f$ is Lipschitz, then you can assume it has a Lipschitz constant as large as you like; in particular, it is safe to assume $K>1$.