Moment conditions for MS GARCH (Haas et al.)

I am doing some research on MS GARCH, specifically the model proposed by Haas et al., however I am really stuck on the moment conditions derived in the appendix of the paper.

• Right before equation ($34$) I am unsure why do we have to make use of the iid random variables. Also I am not quite following the reasoning how equations $34$ are derived, and thus how we can write the conditional expectation as equation ($35$). I know that we can relate $$\mathbb{E}[\boldsymbol{\sigma}_n \mid \mathcal{F}_{n-\tau-1},\boldsymbol{\pi}_{n-\tau-1}]=(\boldsymbol{1}_k' \otimes \boldsymbol{I}_k)\boldsymbol{M}^\tau \boldsymbol{\Pi}_1+(\boldsymbol{1}_k' \otimes \boldsymbol{I}_k)\sum\limits_{i=0}^{\tau-1}\boldsymbol{M}^i\boldsymbol{A}_{n-1-i}^1$$ to $$\mathbb{E}(\boldsymbol{\sigma_{n+1}} \mid \Delta_{n-1}=j, \boldsymbol{\sigma}_n)=\boldsymbol{\alpha_0}+\sum\limits_{i=1}^k \boldsymbol{M}_{ji}\boldsymbol{\sigma_n}$$ however the conditional in these 2 instances is not the same.
• Can someone direct me from where exactly was this equation extracted as I can't seem to recognize it from Karlin and Taylor. $$\pi_{y-\tau-i}=\pi_\infty + \delta^{\tau-1-i}(\pi_{t-\tau}-\pi_\infty)$$
• How are we generalizing for k>2 using perron's formula?