Let $\zeta$ be a primitive $n$-th root of unity. Then $$\prod_{j=1}^{n-1}(x-\zeta^j)=\frac{x^n-1}{x-1}=1+x+\cdots+x^{n-1}$$ and hence $$\sigma_k=\sum_{1\le i_1<\cdots<i_k\le n-1}\zeta^{i_1+\cdots+i_k}=1$$ for all $k=1,\ldots,n-1$.
Observe that \begin{align*}\mathrm{per}[1-\zeta^{j+k}]_{1\le j,k\le n-1}=&\sum_{\tau\in S_{n-1}}\ \prod_{j=1}^{n-1}(1-\zeta^{j+\tau(j)}) \\=&\sum_{\tau\in S_{n-1}}1+\sum_{\tau\in S_{n-1}}\ \sum_{\emptyset\not=J\subseteq\{1,\ldots,n-1\}}(-1)^{|J|}\zeta^{\sum_{j\in J}\ (j+\tau(j))} \\=&(n-1)!+\sum_{\emptyset \not=J\subseteq \{1,\ldots,n-1\}}(-1)^{|J|}\zeta^{\sum_{j\in J}j}\sum_{\tau\in S_{n-1}}\zeta^{\sum_{j\in J}\ \tau(j)}. \end{align*} For $\emptyset \not=J\subseteq\{1,\ldots,n-1\}$, clearly \begin{align*}&\sum_{\tau\in S_{n-1}}\zeta^{\sum_{j\in J}\ \tau(j)} \\=&\sum_{1\le i_1<\cdots<i_{|J|}\le n-1}\zeta^{i_1+\cdots+i_{|J|}}\sum_{\tau\in S_{n-1}\atop\{\tau(j):\ j\in J\} =\{i_1,\ldots,i_{|J|}\}}1 \\=&|J|!(n-1-|J|)!\sigma_{|J|}=|J|!(n-1-|J|)!. \end{align*} Therefore \begin{align*}\mathrm{per}[1-\zeta^{j+k}]_{1\le j,k\le n-1}=&(n-1)!+\sum_{\emptyset\not=J\subseteq\{1,\ldots,n-1\}}(-1)^{|J|}|J|!(n-1-|J|)!\zeta^{\sum_{j\in J}\ j} \\=&(n-1)!\sum_{k=0}^{n-1}\frac{(-1)^k}{\binom{n-1}k}=(1-(-1)^n)\frac{n!}{n+1}. \end{align*} Then the conjectural identity $(2)$ follows from this since $$\sin\pi\frac{j+k}n=\frac{e^{-\pi i(j+k)/n}}{2i}\left(e^{2\pi i(j+k)/n}-1\right).$$
The identities $(3)$ and $(4)$ can be proved similarly. Actually these ideas are slight modification of my way to establish Theorems 1.1 and 1.2 in my preprint Arithmetic properties of some permanents availabel from http://arxiv.org/abs/2108.07723.