$i,ii,iii)$ Define $x_t$ by $x_t=(S_t^1)^2$. Ito's formula give us
\begin{align}
& dx_t=d(S_t^1)^2=2S_t^1dS_t+\frac{1}{2}(2)d[S_t^1,S_t^1](t)\\
& dx_t=d(S_t^1)^2=2S_t^1(\frac{1}{S_t^1}+dW_t)+dW_tdW_t=3dt+2S_t^1dW_t\\
\end{align}
then
\begin{align}
& dx_t=3\ dt+2\sqrt x_t\ dW_t\\
\end{align}
The following portfolio is constructed:
we buy a bond of dollar value $V_1$ with maturity $T_1$ and sell another bond of dollar value $V_2$ with maturity $T_2$. The portfolio value $\Pi$ is given by
\begin{align}
\Pi=V_1-V_2
\end{align}
where the $t$ subscripts are omitted for convenience. Assuming the portfolio is
self-financing, the change in portfolio value is
\begin{align}
d\,\Pi=dV_1-dV_2
\end{align}
The strategy is similar to that for the Black-Scholes case. I apply Ito’s lemma to obtain the processes for $V_1$ , which allows us to find the process for $\Pi$ .To form the hedging portfolio, first apply Ito’s lemma to the value of the derivative,$V_1=V_1(x_t,t)$ We must differentiate $V_1$ with respect to the variables $t$ and $x$,The result is that $dV_1$ follows the process
\begin{align}
dV_1=\frac{\partial V_1}{\partial t}dt+\frac{\partial V_1}{\partial x}dx+\frac{1}{2}\frac{\partial^2 V_1}{\partial x^2}d[x,x](t)
\end{align}
in other words
\begin{align}
dV_1=(\frac{\partial V_1}{\partial t}+3\frac{\partial V_1}{\partial x}+2x_t\frac{\partial^2 V_1}{\partial x^2})dt+2\sqrt x_t \frac{\partial V_1}{\partial x}dW_t
\end{align}
I have use the fact that $d[x,x](t)=(2\sqrt x_t\ dW_t)(2\sqrt x_t\ dW_t)=4x_t dt$ , $dt\,dW_t=0$ and $dt\,dt=0$. Then portfolio value can be written
\begin{align}
&d\,\Pi=(\frac{\partial V_1}{\partial t}+3\frac{\partial V_1}{\partial x}+2x_t\frac{\partial^2 V_1}{\partial x^2})dt+2\sqrt x_t \frac{\partial V_1}{\partial x}dW_t\\
&\,\,\,\,\,\,\,\,\,\,-(\frac{\partial V_2}{\partial t}+3\frac{\partial V_2}{\partial x}+2x_t\frac{\partial^2 V_2}{\partial x^2})dt-2\sqrt x_t \frac{\partial V_2}{\partial x}dW_t\\
\end{align}
In order for the portfolio to be hedged against movements Wiener process last two terms in this equation must be zero. This implies that the hedge parameters must be
\begin{align}
\frac{\partial V_1}{\partial x}=\frac{\partial V_2}{\partial x}
\end{align}
then
\begin{align}
&d\,\Pi=(\frac{\partial V_1}{\partial t}+2x_t\frac{\partial^2 V_1}{\partial x^2})dt\\
&\,\,\,\,\,\,\,\,\,\,-(\frac{\partial V_2}{\partial t}+2x_t\frac{\partial^2 V_2}{\partial x^2})dt\\
\end{align}
The condition that the portfolio earn the risk-free rate, $S^0=1$, implies that the change in portfolio value is $d\,\Pi=S^0\,\Pi dt$
\begin{align}
d\,\Pi=S^0\,\Pi\,dt=\Pi\,dt=(V_1-V_2)dt
\end{align}
By combination these equations we have
\begin{align}
\frac{\partial V_1}{\partial t}+2x_t\frac{\partial^2 V_1}{\partial x^2}-V_1=\frac{\partial V_2}{\partial t}+2x_t\frac{\partial^2 V_2}{\partial x^2}-V2
\end{align}
The above relation is valid for arbitrary maturity dates $T_1$ and $T_2$, so this equation should be independent of maturity $T$.Then
\begin{align}
\frac{\partial V}{\partial t}+2x_t\frac{\partial^2 V}{\partial x^2}-V=\lambda
\end{align}
