For the last question. We assume that
\begin{align*}
S_t = S_0 e^{(r-q-\frac{1}{2}\sigma^2)t + \sigma W_t},
\end{align*}
where $W$ is a standard Brownian motion, $r$ is the interest rate, $q$ is the dividend yield, and $\sigma$ is the volatility.
Then,
\begin{align*}
X_{u+a}-X_a &= (r-q-\frac{1}{2}\sigma^2)a + \sigma(W_{u+a}-W_u)\\
&\sim (r-q-\frac{1}{2}\sigma^2)a + \sigma W_a\\
&= X_a.
\end{align*}
For the forward start option, note that
\begin{align*}
S_T/S_t &= e^{(r-q-\frac{1}{2}\sigma^2)(T-t) + \sigma (W_T- W_t)}\\
&= e^{(r-q-\frac{1}{2}\sigma^2)(T-t) + \sigma \sqrt{T-t}\xi},
\end{align*}
where $\xi$ is a standard normal random variable. Then
\begin{align*}
C(K, t, T) &= e^{-rT} \mathbb{E}\big(S_T/S_t -K)^+ \big)\\
&= e^{-rT}\big[N(d_1) - KN(d_2) \big],
\end{align*}
where $N$ is the cumulative distribution function of a standard normal random variable,
\begin{align*}
d_{1} = \frac{\ln (1/K) + (r-q+ \frac{1}{2}\sigma^2 )(T-t)}{\sigma \sqrt{T-t}},
\end{align*}
and
\begin{align*}
d_{2} = \frac{\ln (1/K) + (r-q- \frac{1}{2}\sigma^2 )(T-t)}{\sigma \sqrt{T-t}}.
\end{align*}
