question regarding econometric testing of the natural rate hypothesis
Equations $11$) and $12$) are not derived, they are assumed. 

Specifically, with eq. $9$) Lucas defines/assumes that the value-expected for the $t+1$ price level is the conditional expectation based on $x_t, x_{t+1},\eta_t$ (plus $\eta_t$ on its own). This is imposing the Rational Expectations Hypothesis.  

He then says "we will seek linear solutions..." which is translated "assume that the functional form of $E(P_{t+1}| x_{t}, x_{t-1}, \eta_{t})$ is linear in its arguments, and then determine (or estimate) the coefficients". Certainly, this is in all likelihood an approximation to the true functional form of this conditional expectation.

So assuming that 

$$E(P_{t+1}| x_{t}, x_{t-1}, \eta_{t}) = \pi_{4} x_{t} +\pi_{5} x_{t-1} + \beta\eta_{t}$$

and inserting this into $9$) we get $12$) with the mapping $\pi_{6} = 1+\beta$.

Inserting $9$) in $10$) we get

$$(1+a) P_{t} - a [\pi_{4} x_{t} + \pi_{5} x_{t-1} + \pi_{6} \eta_{t}] = x_{t}$$

$$\implies P_{t} = \frac{1+a\pi_{4}}{1+a}x_t +\frac{a\pi_{5}}{1+a}x_{t-1}+\frac{a\pi_{6}}{1+a}\eta_{t}$$

which is $11$), after compacting and mapping the above coefficients to the $\pi_1, \pi_2, \pi_3$ ones.