The power series ring $R[[I]]$ for arbitrary index sets $I$

Question

Throughout, $R$ is a commutative ring and $K$ is a field.

In my Algebra I class, today, we covered the power series ring $R[[X]]$. $K[[X]]$ was given as an example of a local ring.

Now, I know about polynomial rings and their universal property: A ring homomorphism $R[X] \to S$ is precisely the same as a ring homomorphism $R \to S$ together with an element $s \in S$. Equivalently, this means that $R[X]$ is the free $R$-algebra on one element.

More generally, for any index set $I$, we can form the polynomial ring $R[I]$, which is given by finite sums of finite products of elements of $R$ and indeterminates $X_i$ for $i \in I$. This satisfies the universal property that a ring homomorphism $R[I] \to S$ is precisely the same as a ring homomorphism $R \to S$ and for every $i \in I$ a choice of element $s_i \in S$. Equivalently, this means that $R[I]$ is the free $R$-algebra on $I$.

Now, I know how to define the power series ring $R[[X]]$. According to Wikipedia, it fulfills the following universal property: It is the initial $R$-algebra equipped with an ideal $J$ such that the $J$-adic topology is complete and and equipped with an element $s \in J$. This means that for any $R$-algebra $S$ and any ideal $J$ of $S$ such that the $J$-adic topology on $S$ is complete as well as any $s \in J$, there is a unique continuous $R$-algebra homomorphism from $R[[X]]$ to $S$ sending $X$ to $s$.

Wikipedia also generalizes this to finitely many variables: $R[[X_1, \ldots, X_n]]$ is the initial $R$-algebra equipped with an ideal $J$ such that the $J$-adic topology is complete and and equipped with elements $s_1, \ldots, s_n \in J$.

It is clear how to generalize this universal property to arbitrary index sets $I$: We want $R[[I]]$ to be the initial $R$-algebra equipped with an ideal $J$ such that the $J$-adic topology is complete and and equipped with elements $s_i \in J$ for $i \in I$. The question is now whether there actually exists an object satisfying this universal property. I've read in the comments of Are there formal power series ring in infinitely many indeterminates that for infinite $I$, there are multiple possibilities for how to define $R[[I]]$. Perhaps one of those works?

Hence my questions:

1. Does there exist an object $R[[I]]$ satisfying the above universal property? If yes, how can we explicitly describe this ring?

2. Is $K[[I]]$ always a local ring with maximal ideal $\langle X_i \mid i \in I \rangle$?