You may easily calculate everything in complex numbers. Denote $m=2n+1$, $w=e^{2\pi i/n}$, $Q_i=A_{i,3}=A_{i+1,2}$ for $i=i,2,\ldots$. We may suppose that $P_{m+i}=P_i$ for $i=1,\ldots,m$ and we have one sequence of $2m$ polygons. Then we have to prove $Q_{2m}=O$ (let $O=0$ be the origin) and that $C_k:=A_{k+m}-A_k$ satisfy $C_{k+1}=-w C_k$. This follows from $Q_k-P_k=(P_k-Q_{k-1})w$, where $Q_0=0$. Dividing by $w^k$ this gives for the sequence $R_k:=(-1)^kQ_k/w^k$ the recurrence $R_k-R_{k-1}=(-1)^k(P_k/w^{k-1}-P_{k-1}/w^k)=:x_k$. We have $x_{m+k}+x_k=0$, thus $R_{2m}=x_1+\ldots+x_{2m}=0$. Also $A_k=(Q_k-P_kw)/(1-w)$, thus $$C_k=A_{k+m}-A_k=\frac{1}{1-w}(Q_{k+m}-Q_k)=(-1)^{k}\frac{w^k}{1-w}(R_k+R_{m+k})= (-1)^{k}\frac{w^k}{1-w}(x_1+\ldots+x_k+x_1+\ldots+x_{m+k})= (-1)^{k}\frac{w^k}{1-w}(x_1+\ldots+x_m),$$ the result follows.

You may easily calculate everything in complex numbers. Denote $m=2n+1$, $w=e^{2\pi i/n}$, $Q_i=A_{i,3}=A_{i+1,2}$ for $i=i,2,\ldots$. We may suppose that $P_{m+i}=P_i$ for $i=1,\ldots,m$ and we have one sequence of $2m$ polygons. Then we have to prove $Q_{2m}=O$ (let $O=0$ be the origin) and that $C_k:=A_{k+m}-A_k$ satisfy $C_{k+1}=-w C_k$. This follows from $Q_k-P_k=(P_k-Q_{k-1})w$, where $Q_0=0$. Dividing by $w^k$ this gives for the sequence $R_k:=(-1)^kQ_k/w^k$ the recurrence $R_k-R_{k-1}=(-1)^kP_k(1+w)/w^k=:x_k$. We have $x_{m+k}+x_k=0$, thus $R_{2m}=x_1+\ldots+x_{2m}=0$. Also $A_k=(Q_k-P_kw)/(1-w)$, thus $$C_k=A_{k+m}-A_k=\frac{1}{1-w}(Q_{k+m}-Q_k)=(-1)^{k}\frac{w^k}{1-w}(R_k+R_{m+k})= (-1)^{k}\frac{w^k}{1-w}(x_1+\ldots+x_k+x_1+\ldots+x_{m+k})= (-1)^{k}\frac{w^k}{1-w}(x_1+\ldots+x_m),$$ the result follows.

As for question 2, it asks when $x_1+\ldots+x_m=0$. Since $O$ is variable, we replace $P_i$ to $P_i-O$ and get the equation $$ \sum_{k=1}^m (-1)^k (P_k-O)/w^k=0 \Leftrightarrow O=\frac{1+w}2\sum_{k=1}^m (-1)^{k-1}P_k w^{-k}. $$