This should be a comment - but it is too long:
Assume that $U = U_1 = U_2$. I want to show that $\mathbb{P}_U ^2 \cong \mathbb{P}_U\times \mathbb{C}$ where $\mathbb{C}$ is the Cohen forcing.
Let $\{ {\alpha^0}_i\}_{i <\omega}, \{ {\alpha^1}_i \}_{i<\omega}$ be the two Prikry sequences.
Set $\{ \gamma_n \}_{n<\omega} = \{ {\alpha^0}_i\}_{i <\omega} \cup \{ {\alpha^1}_i \}_{i<\omega}$ (the Prikry sequence) and $f:\omega \rightarrow P(2)\setminus \{\emptyset\}$, $f(n) = \{ i < 2 | \gamma_n \in \{ {\alpha^i}_m \}_{m < \omega} \}$ (the Cohen real).
This gives us an isomorphism: send conditions from the dense set $( \langle s_0, A\rangle , \langle s_1, A\rangle )$ (the same $A$, $\min A > \max s_0 \cup s_1$) in $\mathbb{P}_U^2$ to $(\langle s_0\cup s_1, A \rangle, f\restriction |s_0 \cup s_1|) \in \mathbb{P}_U \times \mathbb{C}$ (we can calculate $f\restriction |s_0 \cup s_1|$ since we can only add elements in the Prikry sequence above the $\max s_0 \cup s_1$). This is an order preserving bijection between those two posets.
Edit: The answer for Question 1.2 depends on the exact support that you're using:
For finite support - $\mathbb{P}^\omega$ trivially collapses $\kappa$ to $\omega$ - the function that assign to each $n$ the first element in the $n$-th Prikry sequence is onto $\kappa$.
For full support - $\mathbb{P}^\omega$ also collapses $\kappa$: in the generic extension there is a surjection from $(2^{\aleph_0})^V$ to $\kappa$.
Choose a $\omega$-Jonsson function on $\kappa$, i.e. function $f:[\kappa]^\omega \rightarrow \kappa$, such that for every $x\subset \kappa, |x|=\kappa$, $f^{\prime\prime}([x]^\omega) = \kappa$. Let $\{\alpha^j_i \}_{i<\omega}$ be the Prikry sequence added by the $j$ component of $\mathbb{P}^\omega$.
We define a function $g: (\omega^\omega)^V \rightarrow \kappa$ by $g(z) = f(\{\alpha_{z(n)}^n | n < \omega\})$ .
I claim that $g$ is surjective: Let $p = \langle g_i, A_i | i < \omega \rangle \in \mathbb{P}^\omega$ and $\alpha < \kappa$. WLOG, $A=A_i$ for every $i$. Choose $y\in A^\omega$ such that $f(y) = \alpha$ and extends each $g_i$ by the corresponding element of $y$. Since the sequence of lengths of $g_i$ is real from $V$ - the new condition forces $\alpha \in \text{im }g$.
Since we're dealing with Prikry forcing there is at least one more support that we should consider - the Magidor support, namely the conditions are all elements of the form $\langle g_i, A_i |i < \delta\rangle$ such that $\{i<\delta | g_i \neq \emptyset \}$ is finite. In this case (as long as $\delta < \kappa$), we can apply the same idea as above and get that $\mathbb{P}^\delta \cong \mathbb{P}\times \mathbb{D}$ where $\mathbb{D}$ is a forcing that adds a generic function from $\omega$ to the set of all finite, non empty subsets of $\delta$, so as long as $\delta < \kappa$ - $\kappa$ is not collapsed.
When $\delta = \kappa$ this argument doesn't work, so I don't know if $\kappa$ is collapsed by the Magidor power $\mathbb{P}^\kappa$ or not.
--
Remark 1. The answer to question 1.1 is yes, even if $U_1\neq U_2.$
Theorem. Let $U,V$ be normal measures on $\kappa.$ Then forcing with $\mathbb{P}_U\times \mathbb{P}_V$ preserves all cardinals.
Proof. It suffices to consider the case where $U$ is not equal to $V$. So let $A^*\in U$ such that $\kappa-A^*\in V.$ Let $W=\{ X\subset\kappa: X\cap A^*\in U, X\cap (\kappa-A^*)\in V \}.$ It is easily seen that $W$ is $\kappa-$complete filter on $\kappa$ which is Rowbottom (for any $f: [D]^{<\omega}\to \lambda<\kappa, D\in W$, there is $E\in W, E\subset D$ such that $card(f'' [E]^{<\omega}) \leq \omega$ ). So we can define $\mathbb{P}_W$ and by Devlin's paper "Some Remarks on Changing Cofinalities"1, forcing with $\mathbb{P}_W$ preserves cardinals. As above argument, we have a forcing isomorphism from the dense subset $\{((s, A)(t, B))\in \mathbb{P}_U\times \mathbb{P}_V: A\subset A^*, B\subset \kappa-A^*, max(s\cup t)< min(A), min(B) \}$ of $\mathbb{P}_U\times \mathbb{P}_V$ to $\mathbb{P}_W\times \mathbb{C}$ given by $((s, A)(t, B))\to ((s\cup t, A\cup B), f\restriction |s\cup t|),$ where $f$ is defined as above argument. The result follows.
- Keith J. Devlin, Some remarks on changing cofinalities, J. Symbolic Logic 39 (1974), 27--30.
Remark 2. $\kappa$ is collapsed by the Magidor power $\mathbb{P}^\kappa$, by the following argument:
For any $\delta<\kappa,$ we may factor $\mathbb{P}^\kappa$ as $\mathbb{P}^\delta \times \mathbb{P}^{\kappa -\delta}$, so by the above argumets we can conclude that all cardinals $<\kappa$ are collapsed, so $\kappa$ is collapsed since it is singular in the extension.
Cantor's back-and-forth theorem quoted in the OP has a model-theoretic generalization.
If two $\tau$--structures $A$ and $B$ in a vocabulary $\tau$ are partially isomorphic, then there is a forcing extension in which they are isomorphic.
If $A$ and $B$ are partially isomorphic countable $\tau$--structures in a countable vocabulary $\tau$, then $A \simeq B$.
In particular, if $A$ and $B$ are $L_{\infty \omega}$--equivalent and countable, then $A \simeq B$.
Several interesting results first proved by forcing are also listed in the answers to the question: https://mathoverflow.net/a/53887/57583
Best Answer
The key here is that you don't have to force beyond $\mathrm{Add}(\kappa,\kappa^+)$ in order to lift the embedding.
Suppose $G \subseteq \mathbb P$ is generic and $H \subseteq \mathrm{Add}(\kappa,\kappa^+)$ is generic over $V[G]$. Since we have GCH, $j(\kappa) < \kappa^{++}$. We also have $M^\kappa \subseteq M$, and this is preserved by the forcing so that $M[G*H]^\kappa \subseteq M[G*H]$ in $V[G*H]$.
Now the tail of the iteration, $j(\mathbb P)/(G*H)$ is $\kappa^+$-closed and of size $j(\kappa)$ and with the $j(\kappa)$-c.c., but from the perspective of $V[G*H]$, it has only $\kappa^+$-many maximal antichains. Thus we may inductively build a filter $F$ for this that is generic over $M[G*H]$, and lift the embedding to $j : V[G] \to M[G']$ (where $G' = G*H*F$).
So by forcing with $\mathbb P * \dot{\mathrm{Add}}(\kappa,\kappa^+)$, we get an elementary embedding lifting the ultrapower embedding. Applying Foreman's Duality Theorem (proof given as 2.12 here), with $I$ being the dual of $\mathscr U$ in that notation, there is in $V[G]$ a normal ideal $J$ on $\kappa$ such that $P(\kappa)/J$ is isomorphic to what I called $(j(\mathbb P)/\dot K)/G$ there. What is $\dot K$? In this case, it is the dual ideal (in the Boolean completion) to the filter of elements forced to be in $G*H*F$, and $j(\mathbb P)/\dot K \sim \mathbb P * \dot{\mathrm{Add}}(\kappa,\kappa^+)$, and $(j(\mathbb P)/\dot K)/G \sim\dot{\mathrm{Add}}(\kappa,\kappa^+)$.