I will answer the first question, I believe the second one can be found in the book's appendix.
"Price stickiness" is defined with respect to the optimal price level for the period (here denoted by a star) or equivalently in inflation terms. We do not have price stickiness if the current inflation equals the optimal inflation:
$$\pi_t = \pi^*_t$$
Looking at eq. $(9.26)$, written
$$\pi_t = \alpha\pi^*_t + \beta \pi^e_{t+1}$$
to allow for any expectations formation hypothesis (we will need this flexibility), here are the only two scenarios under which we do not have price stickiness, while also Wickens' "general formulation" holds:
A. Assume $\beta = 1-\alpha$ and $\pi^e_{t+1} = \pi^*_t$. Then we get $\pi_t = \pi^*_t$. Now, $\pi^e_{t+1} = \pi^*_t$ can be observed, it may be the case under Rational Expectations to expect next period inflation to equal the current optimal inflation. But assuming $\beta = 1-\alpha$ is a very special and uninteresting case.
B. Impose immediate and full adjustment, $\pi_t = \pi^*_t$, while respecting equation $(9.26)$: we get a very specific rule for expectations formation:
$$\pi^*_t = \alpha\pi^*_t + \beta \pi^e_{t+1} \implies \pi^e_{t+1} = \frac {1-\alpha}{\beta}\pi^*_t$$
This is clearly an ad hoc expectations formation assumption.
It follows that in any other case, eq. $(9.26)$ implies that $\pi_t \neq \pi^*_t$, i.e. we have partial adjustment. It is perhaps more illuminating to perform the above "what ifs" using eq. $(9.27)$ of the book that is written in terms of the price level (starting in long-run equilibrium, i.e. assuming $p_{t+1} = p^*_{t+1}$).
where $p^*_t$ is the optimal price at time $t$.
The three theories he is referring to are: Taylor Model of Overlapping contracts, Calvo pricing model, and Optimal Dynamic adjustment. He says that all can be encompassed in this reformulation. The author then proceeds to show why this general formulation of inflation exhibits price stickiness, and this is the part which I don't understand what he does, specifically what he calls the partial adjustment model:
