Q1: There is no such partition. Let $\langle X_n \rangle$ be a countable partition of $\mathbb{R}.$ We will construct $n,$ an open interval $I,$ and an injection $g: \omega \rightarrow I \cap X_n$ with $\text{rng}(g)$ dense in $I.$ We recursively construct $\langle (p_{\alpha}, S_{\alpha}): \alpha<\omega_1 \rangle,$ with $p_{\alpha} \in \mathbb{R}$ and $S_{\alpha}=\{p_{\xi}: \xi<\alpha\}.$ At stage $\alpha,$ if for some $n,$ $X_n \cap S_{\alpha}$ is somewhere dense, set $p_{\alpha}=0.$ Otherwise, follow the procedure of the proof of the Baire category theorem (with respect to a basis $\langle U_k \rangle$ of rational open intervals) to construct
$$p_{\alpha} \in \mathbb{R} \setminus \left ( 0 \cup \bigcup_{n<\omega} \overline{X_n \cap S_{\alpha}} \right ).$$
Consider the limit ordinal $\beta = \{\alpha<\omega_1: p_{\alpha} \neq 0\}.$ Define a surjective partial map $h: \omega^2 \rightharpoonup \beta$ by setting $h(k, n) = \alpha$ if $\alpha$ is least such that $p_{\alpha} \in U_k \cap X_n.$
In particular, $\beta$ is countable, and least such that $p_{\beta}=0.$ Let $(n, k)$ be lexicographically least such that $X_n \cap S_{\beta}$ is dense in $U_k.$ Then set $I=U_k$ and use $h$ to construct bijective $g: \omega \rightarrow X_n \cap S_{\beta} \cap I.$
Q2: Indeed, in the Cohen model there is a partition of $\mathbb{R}$ into two sets of strictly smaller cardinality. The Bernstein set $B$ in Theorem 1.7 here works, by a minor adjustment of their argument. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be injective, and $b \in [A]^{<\omega}$ the minimal parameter from which $f$ is constructible. Then carry out their argument in terms of $f$ and $b$ rather than $T$ and $a$ to show that $B \cap \text{rng}(f)$ and $B \setminus \text{rng}(f)$ are nonempty.
Q3&4: Consistently, $\mathbb{R}$ is a Dedekind finite union of pairs. Start in $L,$ let $G=\langle c_{\alpha}: \alpha<\omega_1 \rangle$ be an $L$-generic sequence of Cohen reals. In $L[G],$ let $A=\{c_{\alpha}: \alpha<\omega_1\}$ and $$R=\bigcup_{a \in [A]^{<\omega}} \mathbb{R}^{L[a]}.$$
Let $\pi=(\pi_1, \pi_2) \in L$ be the standard bijection from $\omega_1 \rightarrow \omega_1^2.$ We partition $A$ by letting $A_{\alpha}=\{c_{\xi}: \xi \in \pi_1^{-1}(\alpha)\}.$ Let $$M=L(R, \langle A_{\alpha}: \alpha<\omega_1\rangle).$$ We will show $M$ has our desired property.
Claim: $R=\mathbb{R}^M.$
Proof of claim: Fix $r \in \mathbb{R}^M.$ There is $\varphi,$ an ordinal $\gamma,$ and $a=\{c_{\alpha_0}< \ldots<c_{\alpha_i}\}\in [A]^{<\omega}$ such that, letting $G'=G \restriction \omega_1 \setminus \{\alpha_j\},$ we have for all $n<\omega$ that
\begin{align*}n \in r &\Leftrightarrow M \models \varphi(n, \gamma, c_{\alpha_0}, \ldots, c_{\alpha_i}, \langle A_{\alpha} \rangle)
\\&\Leftrightarrow L[G] =L[c_{\alpha_0}, \ldots, c_{\alpha_i}][G'] \models
\varphi^{L(R, \langle A_{\alpha} \rangle)}(n, \gamma, c_{\alpha_0}, \ldots, c_{\alpha_i}, \langle A_{\alpha} \rangle)
\\&\Leftrightarrow L[c_{\alpha_0}, \ldots, c_{\alpha_i}] \models \text{Add}(\omega, \omega_1 \setminus \{\alpha_j\}) \Vdash \varphi^{L(R, \langle A_{\alpha} \rangle)}(n, \gamma, c_{\alpha_0}, \ldots, c_{\alpha_i}, \langle A_{\alpha} \rangle),
\end{align*}
the last forward implication justified by a standard Cohen homogeneity argument, using the invariance of $R$ and $\langle A_{\alpha} \rangle$ under
$\pi_1$-preserving permutations of $\omega_1.$
Thus, $r \in \mathbb{R}^{L[c_{\alpha_0}, \ldots, c_{\alpha_i}]} \subset R.$ This proves the claim. $\square$
Suppose some $r \in R$ codes an injective sequence $\langle c_n: n<\omega \rangle \subset A.$ Then $\{c_n\} \subset L[a]$ for some $a \in [A]^{<\omega},$ contradicting the mutual genericity of the Cohen reals.
Thus, $A$ is Dedekind finite in $M,$ which implies $[A]^{<\omega} \subset [\mathbb{R}]^{<\omega} \equiv \mathbb{R}$ is also Dedekind finite. We define a surjection $f: [A]^{<\omega} \setminus \{\emptyset\} \rightarrow R$ by sending $a$ with $\max a \in A_{\alpha}$ to the $\alpha\text{th}$ real in $L[a \setminus \{\max a\}].$ Then
$$R=\bigcup_{a \in \text{dom} f \subset \mathbb{R}} \{a, f(a)\chi_{f^{-1}(\mathbb{R} \setminus \text{dom} f)}(a)\}$$
is as desired.
