Solve the integral $S_k = (-1)^k \int_0^1 (\log(\sin \pi x))^k dx$

My nephew asked me this, so I suggested him to sign up here. But anyways this question I was trying to solve myself. I got part of the solution. Let me know the rest.

$(1)$ Solve the integral defined as

$\displaystyle{S_k = (-1)^k \int_0^1 (\log(\sin \pi x))^k dx}$ and show

$(2)$ $\displaystyle{S_k = \frac{(-1)^k}{\sqrt\pi 2^k} \frac{d^k}{d\alpha^k} \frac{\Gamma(\alpha+\frac{1}{2})}{\Gamma(\alpha+1)}}$ with $\alpha=0$.

$(3)$ Show that $\displaystyle{S_4 = \frac{19 \pi^4}{240}+\frac{1}{2} \pi^2 \log^2 2 + \log^4 2 + 6 \log 2 \, \zeta(3)}$

$(4)$ Show the following: $\displaystyle{\int_0^1 \log \log \left(\frac{1}{x}\right) \frac{dx}{1+x^2} = \frac{\pi}{2}\log \left(\sqrt{2\pi} \Gamma\left(\frac{3}{4}\right) / \Gamma\left(\frac{1}{4}\right)\right)}$

For $(3)$, if I substitute $y=\pi x$, I can transform this to a well known log-sine function as described here at Wolfram, and so showing $\displaystyle{S_4 = \frac{19 \pi^4}{240}+\frac{1}{2} \pi^2 \log^2 2 + \log^4 2 + 6 \log 2 \, \zeta(3)}$ is not hard.

I would like suggestions to read or partial solutions (not complete solutions).

(NOTE: I also noted that someone have asked a similar question for $k=2$ here)

Problem (2): Use generating series. Consider the function $$f(z)=\sum_{k=0}^{\infty}S_{k}\frac{z^{k}}{k!}.$$ Then $$f(z)=\int_0^1 \left(\sum_{k=0}^\infty (-1)^k \log^k(\sin(\pi x)) z^k \right)dx=\int_{0}^{1}\frac{1}{\sin\left(\pi x\right)^{z}}dx=\frac{\Gamma\left(\frac{1-z}{2}\right)}{\sqrt{\pi}\Gamma\left(1-\frac{z}{2}\right)}.$$ The last equality follows from an identity of the Beta Function and then applying the Duplication Formula. From here, differentiating and plugging in $z=0$ allows you to conclude (2).

(You said you wanted few details)

As for the the problem (2) we have $$ \begin{align} S_k&=(-1)^k \int\limits_0^1\log^k(\sin\pi x)dx\\ &=\frac{(-1)^k}{\pi} \int\limits_0^\pi\log^k(\sin y)dy\\ &=\frac{(-1)^k}{\pi} \int\limits_0^{\pi/2}\log^k(\sin y)dy+ \frac{(-1)^k}{\pi} \int\limits_{\pi/2}^\pi\log^k(\sin y)dy\\ &=\frac{(-1)^k}{\pi} \int\limits_0^{\pi/2}\log^k(\sin y)dy+ \frac{(-1)^k}{\pi} \int\limits_{0}^{\pi/2}\log^k(\sin y)dy\\ &=\frac{2(-1)^k}{\pi} \int\limits_0^{\pi/2}\log^k(\sin y)dy\\ &=\frac{2(-1)^k}{\pi} \int\limits_0^{1}\frac{\log^k(t)}{\sqrt{1-t^2}}dt \end{align} $$ Now following Sasha's idea consider integral $$ I(\alpha)= \int\limits_0^1\frac{t^{2\alpha}}{\sqrt{1-t^2}}dt= \frac{1}{2}B\left(\frac{1}{2},\frac{2\alpha+1}{2}\right)= \frac{\sqrt{\pi}}{2}\frac{\Gamma\left(\frac{2\alpha+1}{2}\right)}{\Gamma\left(\frac{2\alpha+2}{2}\right)} $$ Note that $$ I^{(k)}(\alpha)=\int\limits_0^1\frac{t^{2\alpha}2^k\log^k(t)}{\sqrt{1-t^2}}dt $$ so $$ S_k= \frac{2(-1)^k}{\pi} \int\limits_0^{1}\frac{\log^k(t)}{\sqrt{1-t^2}}dt= \frac{2(-1)^k}{\pi} \frac{1}{2^k}I^{(k)}(0)= \frac{(-1)^k}{2^k\sqrt{\pi}}\frac{d^k}{d\alpha^k}\left(\frac{\Gamma\left(\alpha+\frac{1}{2}\right)}{\Gamma\left(\alpha+1\right)}\right)_{\alpha=0} $$