Regarding the first question: Let $\mathbb F$ be a field (not necessarily $\mathbb R$ or $\mathbb C$), and let $m,n$ be nonnegative integers. Given an involutory field automorphism $a:\mathbb F\to\mathbb F$ (think complex conjugation), we can define a sort of "conjugate transpose" $\mathbb F^{m\times n}\to\mathbb F^{n\times m}$ via $A\mapsto A^*:=\left(a(A_{j,i})\right)_{i\in n,j\in m}$. This "conjugate transpose" lets us define what it means for a matrix $X\in\mathbb F^{n\times m}$ to be a pseudoinverse of $A.$ It can then be shown that every $A\in\mathbb F^{m\times n}$ has at most one pseudoinverse $X\in\mathbb F^{n\times m}$ (the usual proof carries over). The topic of existence is more delicate; it can be shown that an arbitrary $A\in\mathbb F^{m\times n}$ has a pseudoinverse belonging to $\mathbb F^{n\times m}$ if and only if we have the equalities $\text{rank}(A^*A)=\text{rank }A=\text{rank}(AA^*)$ [Bot13].
Now, consider the case where we have two fields $\mathbb E,\mathbb F$, and an involutory field automorphism $a:\mathbb F\to\mathbb F,$ where $\mathbb E\subseteq\mathbb F$ and $a|_{\mathbb E}:\mathbb E\to\mathbb E$ has range $\text{range}(a|_{\mathbb E})\subseteq\mathbb E.$ It is not difficult to see that $a|_{\mathbb E}$ will be an involutory field automorphism on $\mathbb E$, so, given two nonnegative integers $m,n$, we can apply the above theory to $\mathbb E^{m\times n}:$ Any $A\in\mathbb E^{m\times n}$ has a pseudoinverse belonging to $\mathbb E^{n\times m}$ iff $\text{rank}_{\mathbb E}(A^*A)=\text{rank}_{\mathbb E}A=\text{rank}_{\mathbb E}(AA^*).$ This string of equalities is in turn equivalent to $\text{rank}_{\mathbb F}(A^*A)=\text{rank}_{\mathbb F}A=\text{rank}_{\mathbb F}(AA^*)$ (rank does not change under field extension). This second string of equalities is equivalent to $A$ having a pseudoinverse belonging to $\mathbb F^{n\times m}$. Thus, $A$ has a pseudoinverse belonging to $\mathbb F^{n\times m}$ iff $A$ has a pseudoinverse belonging to $\mathbb E^{n\times m}$. (One direction is almost obvious, from the set inclusion $\mathbb E^{n\times m}\subseteq \mathbb F^{n\times m}$, but the converse might have been not obvious.)
Now, let us specialize to the case where $\mathbb F=\mathbb C$, $a=(\bar z)_{z\in\mathbb C}$ is complex conjugation, and $\mathbb E$ is any subfield of $\mathbb C$ such that $a|_{\mathbb E}:\mathbb E\to\mathbb E$. (Examples: $\mathbb C$ itself; the algebraic numbers $\mathbb A$; or any subfield of $\mathbb R$, such as $\mathbb R$ itself, $\mathbb Q$, or $\mathbb Q(\sqrt2)$.) We know that every $A\in\mathbb E^{m\times n}\subseteq\mathbb C^{m\times n}$ has a pseudoinverse, which we denote $A^+$, belonging to $\mathbb C^{n\times m}.$ Thus, from the above paragraph, $A$ has a pseudoinverse in $\mathbb E^{n\times m}\subseteq\mathbb C^{n\times m}$, and so from uniqueness in $\mathbb C^{n\times m}$ we have $A^+\in\mathbb E^{n\times m}$. Thus, the answer is yes.
In fact, for any $A\in\mathbb Z^{m\times n}$, the least common denominator of the image of $A^+$,
$$\ell:=\min\{d\in\mathbb Z_{\ge1}:\forall i\in m~\forall j\in n\quad d(A^+)_{i,j}\in\mathbb Z\}$$ will divide (in $\mathbb Z$) $(\text{vol}A)^2$, where we define the volume
$$\text{vol}~A:=\sqrt{\sum_{I,J}\det(A|_{I\times J})^2},$$
where the summation ranges over all cardinality-$r$ subsets $I,J\subseteq A$, where $r=\text{rank}~A$ [BhR02 p.81 Theorem 6.7, BouVre20]. (If $A$ is square and invertible, then $\text{vol}~A=|\det A|$.) In fact, based on a few examples I computed (with noninvertible integer-valued matrices whose nonzero entries are relatively prime), it seems we "often" have $\ell=(\text{vol}~A)^2$. For instance, in the example $[1;2]^+=5^{-1}[1,2]$ we have $\ell=5$.
Regarding the fourth question: We also have a necessary and sufficient condition when the entries of $A$ itself are integers: if $A\in\mathbb Z^{m\times n}$, then $A^+\in\mathbb Z^{n\times m}$ iff $\text{vol}~A=1$ [BhR02 Theorem 6.7]. As for arbitrary $A\in\mathbb Q^{m\times n}$, I'm unsure.
[BhR02] Bhaskara Rao, K. (2002). Theory of generalized inverses over commutative rings. CRC Press.
[Bot13] Botha, J. D. (2013). Matrices over finite fields. In: Hogben, L. (ed.). Handbook of Linear Algebra (2nd ed.). CRC Press. Section 31.5.
[BouVre20] Bouman, N. J., and de Vreede, N. (2020). A practical approach to the secure computation of the Moore–Penrose pseudoinverse over the rationals. In International Conference on Applied Cryptography and Network Security (pp. 398-417). Springer, Cham.
