Solving rational expectations model - Sims form
I think I have managed to solve it. However, not the way I was initially hoping. I simplified the stacked matrices using the given conditions and some assumptions. Here is my solution:

Eq. (3) I write as $\pi_{t+1} = \alpha \beta \pi_{t} + \beta \theta_{t} + \eta_{t+1}$

Forwarding equation (4) one period and arranging it in terms on $b_{t+1} = -\varphi_{1} {\pi_{t+1}} - \varphi_{2}{\pi_{t}}  + (\beta^{-1}- \gamma)\widetilde{b_{t}} - \varphi_{3} \theta_{t+1} \ - \psi_{t+1} - \varphi_{4} \theta_{t} = 0$

$\pi_{t+1}$ simplified by eq. (3)

$\theta_{t+1} = \rho_{1} \theta_{t} + \epsilon_{1,t+1}$

$\psi_{t+1}$ can be written as its AR(1) formulation as well. Putting the four equations in matrix form results in:



Which is what I was after. However, this made me wonder what was wrong with the method I was trying above. Looking at the top corner of the coefficient matrix (i.e the 4x4 matrix in the top left corner) it is clearly $\Gamma_{0}^{-1} \Gamma_{1}$. But the rest of the matrix I am unsure of. 

And question (2) remains, once I have solved for the variables of interest, will I be ready to make IRF's?