Computation of a Hankl determinant

The goal of this problem is to compute the determinant $\displaystyle\left(\frac{1}{i+j-1}\right)_{1\leqslant i,j\leqslant n}$ thanks to orthogonal polynomials.
One recalls that there exists some basis $(L_n)_n$ of monic polynomials from $\mathbb{R}[X]$ such that $L_n(-X)=(-1)^nL_n(X)$ and
$$\forall n,m\in\mathbb{N}, \int_{-1}^1 L_n(x)\,L_m(x)\,\mathrm{d}x=h_n'\,\delta_{n,m}$$where $\delta$ is the Kronecker symbol and $\,h_n':=\displaystyle\frac{2^{2n+1}}{2n+1}\frac{(n)!^4}{(2n)!^2}$. The polynomial $L_n$ is of degree $n$ and is called the $n$-th Legendre’s polynomial.

1. Show that there exists another sequence $(P_n)_n$ of monic real polynomial such that $\mathrm{deg}(P_n)=n$ and
   $$\int_{0}^1 P_n(x)\,P_m(x)\,\mathrm{d}x=h_n\,\delta_{n,m}$$where $h_n:=\displaystyle\frac{(n)!^4}{(2n+1)!(2n)!}$. They are the polynomials we will focus on in the following.

Now we define the following Gram matrix:
$$G:=\left(\int_{0}^{1}x^{i-1}x^{j-1}\,\mathrm{d}x\right)_{1\leqslant i,j\leqslant n}$$

2. What is the relation between $H$ and $G$ ?

3. Prove that
   $$\mathrm{det}(G)=\mathrm{det}\left(\left(\int_{0}^{1}Q_{i-1}(x)\,Q_{j-1}(x)\,\mathrm{d}x\right)_{1\leqslant i,j\leqslant n}\right)$$for any sequence $(Q_j)_j$ of monic polynomials such that $\mathrm{deg}(Q_j)=j$.
   In particular this formula is true for $Q_j=P_j$ where $P_j$ are the polynomials we defined above.

4. Use questions 1), 2) and 3) to finally find the formula :
   $$\displaystyle\mathrm{det}(H)=\prod_{i=0}^{n-1}h_i=\prod_{k=0}^{n-1}\frac{(k!)^{4}}{(2k)!(2k+1)!}.$$

and one can check that this formula is equivalent the one that can be found at the end of the exercice Cauchy determinantal formula.

This method, meaning the use of orthogonal polynomial theory, is in particular very solid in order to compute all (or almost) possible determinants of Hankl matrices, meaning $(\mu_{i+j})_{1\leqslant i,j\leqslant n}$ where $(\mu_n)_n$ represents the sequence of moments of a particular scalar product. See reference below (in french) for more details.