I am still not very familiar with these kinds of proofs, but from what I understood:
1) This inclusion is actually a condition which defines in itself what a hiding key is. That is to say that this paragraph does not say that the inclusion is ensured, but that we should make sure that it is, if we want a perfectly hiding key. However, you may find sets of $u_i$'s that verify this property in the three different instantiations. If you use the short version of the paper where they do not appear, look here instead.
In the second (SXDH) instantiation for instance, to commit in $G_1$, you get $u_1=(\mathcal{P}_1,\alpha\mathcal{P}_1)$ and either $u_2=tu_1$ for a binding key, or $u_2=tu_1-(\mathcal{O},\mathcal{P}_1)$ for a hiding key ($t$ and $\alpha$ are chosen at random). In this last case, $\iota(G_1)\subset \langle u_1,u_2\rangle$ because for $g\in G_1$, you can write $g=\mathcal{P}_1^n$ and then you have $\iota(g)=(\mathcal{O},\mathcal{P}_1^n)=(tu_1-u_2)^n\in\langle u_1,u_2\rangle$. Note that you don't have this with the binding parameters.
2) You are right, they probably meant to write $p\circ\iota$ instead.