Convergence of infinite products of formal power series

Question

Setting

Let $R$ be a domain. The ring $R[[X]]$ of formal power series is a complete ultrametric space, see this Wikipedia article. According to the same source, "the philosophy of formal power series is [...] to make convergence questions as trivial as they can possibly be."

In this spirit, we have these nice consequences:

1. An infinite sum $\sum_{i=0}^\infty \alpha_i$ of formal power series $\alpha_i\in R$ converges if and only if $\lim_{i\to\infty}\alpha_i = 0$
2. An infinite product $\prod_{i=0}^\infty \alpha_i$ of formal power seriecs $\alpha_i\in R\setminus\{0\}$ converges (which by convention does not allow the limit $0$) if and only if $\lim_{i\to\infty}\alpha_i = 1$.
3. Independence on the order of the summands: For all $\alpha_i\in R[[X]]$ and all permutations $\pi : \mathbb N \to \mathbb N$, we have $$\sum_{i=0}^\infty \alpha_i = \sum_{i=0}^\infty \alpha_{\pi(i)}.$$(The equality sign is to be read as "the left hand side converges if and only if the right hand side converges, and in that case, the limits coincide.")
4. Discrete Fubini: For all $\beta_{i,j}\in R[[X]]$, $$\sum_{i=0}^\infty \sum_{j=0}^\infty \beta_{i,j} = \sum_{j=0}^\infty \sum_{i=0}^\infty \beta_{i,j}.$$ (The equality sign is to be read as "all terms on the the left hand side converge if and only if all terms on the right hand side converge, and in that case, the outer limits coincide.")

A proof for most of these statements can be found in An invitation to formal power series.

Now I'm wondering about the counterparts of 3. and 4. for infinite products.

Question

Let $R$ be a domain. Which of the following statements are true? (formal proof or counterexample)

• $\displaystyle\prod_{i=0}^\infty \alpha_i = \prod_{i=0}^\infty \alpha_{\pi(i)}$ for all $\alpha_i\in R[[X]]\setminus\{0\}$,

• $\displaystyle\prod_{i=0}^\infty \prod_{j=0}^\infty \beta_{i,j} = \prod_{j=0}^\infty \prod_{i=0}^\infty \beta_{i,j}$ for all $\beta_{i,j}\in R[[X]]\setminus\{0\}$.

Addition

I just found this related question. It covers the first part of my question and indicates that the answer should be "yes". Unfortunately, there is no answer.