Ramiro, regarding 2), the set of all points in $I^\kappa$ with at most countably many non-zero coordinates is even Frechet-Urysohn, for every $\kappa$. This is due to Noble (see also exercise 3.10.D in Engelking).

Ramiro, regarding 2), the set of all points in $I^\kappa$ with at most countably many non-zero coordinates is even Frechet-Urysohn, for every $\kappa$. This is due to Noble (see also exercise 3.10.D in Engelking).

Note: Frechet-Urysohn means that given a non-closed set $A$ and a point $x \in \overline{A} \setminus A $, there is a sequence inside $A$ converging to $x$.