Let $R=k[t^3,t^4,t^5]$ so $\tilde R=k[t]$. I claim that for this ring any immediate ring $R\subsetneq R'\subsetneq \tilde R$ is not reflexive.

Let $m=(t^3,t^4,t^5)$ be that maximal ideal of $R$. As the semigroup generated by $(3,4,5)$ contains all numbers from $3$, $mR'\subseteq m\tilde R\subset R$. It follows that $Hom_R(R',R)$ contains $m$, and since it has to be an ideal of $R$ and can not be $R$, it is $m$.

So now we just need to show that $Hom_R(m,R)\neq R'$. But this is clearly equal to $\tilde R$, as multiplication by $t^i, i\geq 0$ maps $m$ to $R$ and no negative power works.

In general, I think for a non-Gorenstein domain of dimension one, torsion free modules are rarely reflexive.

The answer to Question 1 is yes by (the proof of) Proposition 2.14 in this paper.

The answer to Question 2 is not. Let $R=k[t^3,t^4,t^5]$ so $\tilde R=k[t]$. I claim that for this ring any immediate ring $R\subsetneq R'\subsetneq \tilde R$ is not reflexive.

Let $m=(t^3,t^4,t^5)$ be that maximal ideal of $R$. As the semigroup generated by $(3,4,5)$ contains all numbers from $3$, $mR'\subseteq m\tilde R\subset R$. It follows that $Hom_R(R',R)$ contains $m$, and since it has to be an ideal of $R$ and can not be $R$, it is $m$.

So now we just need to show that $Hom_R(m,R)\neq R'$. But this is clearly equal to $\tilde R$, as multiplication by $t^i, i\geq 0$ maps $m$ to $R$ and no negative power works.

In general, I think for a non-Gorenstein domain of dimension one, torsion free modules are rarely reflexive.

Let $R=k[t^3,t^4,t^5]$ so $\tilde R=k[t]$. I claim that for this ring any immediate ring $R\subsetneq R'\subset \tilde R$ is not reflexive.

Let $m=(t^3,t^4,t^5)$ be that maximal ideal of $R$. As the semigroup generated by $(3,4,5)$ contains all numbers from $3$, $mR'\subseteq m\tilde R\subset R$. It follows that $Hom_R(R',R)$ contains $m$, and since it has to be an ideal of $R$ and can not be $R$, it is $m$.

So now we just need to show that $Hom_R(m,R)\neq R$. But multiplication by $t$ maps $m$ to $R$, and $t$ is not in $R$.

In general, I think for a non-Gorenstein domain of dimension one, torsion free modules are rarely reflexive.

Let $R=k[t^3,t^4,t^5]$ so $\tilde R=k[t]$. I claim that for this ring any immediate ring $R\subsetneq R'\subsetneq \tilde R$ is not reflexive.

Let $m=(t^3,t^4,t^5)$ be that maximal ideal of $R$. As the semigroup generated by $(3,4,5)$ contains all numbers from $3$, $mR'\subseteq m\tilde R\subset R$. It follows that $Hom_R(R',R)$ contains $m$, and since it has to be an ideal of $R$ and can not be $R$, it is $m$.

So now we just need to show that $Hom_R(m,R)\neq R'$. But this is clearly equal to $\tilde R$, as multiplication by $t^i, i\geq 0$ maps $m$ to $R$ and no negative power works.

In general, I think for a non-Gorenstein domain of dimension one, torsion free modules are rarely reflexive.