$\newcommand\R{\Bbb R}\newcommand{\1}{\mathbf{1}}\newcommand{\xx}{\mathbf{x}}\newcommand{\yy}{\mathbf{y}}\newcommand{\uu}{\mathbf{u}}\newcommand{\vv}{\mathbf{v}}\newcommand{\0}{\mathbf{0}}$
Your questions were answered by Remark 3.2 and Lemma 5.1 of this paper. For convenience, here is a reproduction of that, with a few little modifications.
Let us say that a function
$F\colon\R^p\to\R$ is an antiderivative of a function $f\colon\R^p\to\R$ if
\begin{equation*}
F^{(1,\dots,1)}=f;
\end{equation*}
that is, if $F$ is differentiated once with respect to every one of the $p$ arguments of the function $F$, then the result of this $p$-fold partial differentiation is the function $f$. It is assumed that this result does not depend on the order of the arguments with respect to which the partial derivatives are taken.
Here and elsewhere, it suffices to assume that the function $f\colon\R^n\to\R$ is continuous.
Clearly, this notion of an antiderivative is a generalization of the corresponding notion for functions on $\R$.
Remark:
A function $F\colon\R^p\to\R$ is an antiderivative of the function $f$ if and only if one has a representation of the form
\begin{equation*}
F(\xx)=\int_\0^\xx d\yy\, f(\yy)+\sum_{j=1}^p c_j(x_1,\dots,x_{j-1},x_{j+1},\dots,x_p)
\end{equation*}
for all $\xx=(x_1,\dots,x_p)\in\R^p$, where $c_1,\dots,c_p$ are functions on $\R^{p-1}$ such that,
for each $j\in\{1,\dots,p\}$ and all $(x_1,\dots,x_p)\in\R^p$, the mixed partial derivative
$\dfrac{\partial^{p-1}c_j(x_1,\dots,x_{j-1},x_{j+1},\dots,x_p)}{\partial x_1\cdots\partial x_{j-1}\partial x_{j+1}\cdots\partial x_p}$ exists and does not depend on the order of the arguments with respect to which the partial derivatives are taken. The notation $\int_\0^\xx d\yy\, f(\yy)$ is a special case of notation explained in Lemma 1 below.
The "if" part of the above statement is obvious.
The "only if" part of it follows from the multidimensional version of the fundamental theorem of calculus to be given by Lemma 1 below.
In particular, the function $F$ on $\R^p$ given by the condition $F(\xx)=\int_\0^\xx d\yy f(\yy)$ for all $\xx\in\R^p$ is clearly an antiderivative of $f$; thus, there always exists an
antiderivative of the function $f$. $\quad\Box$
The following lemma is a direct multidimensional generalization of the fundamental theorem of calculus (FTC).
Lemma 1:
Let $F$ be any antiderivative of $f$. Take any $\uu=(u_1,\dots,u_p)$ and $\vv=(v_1,\dots,v_p)$ in $\R^p$.
Then
\begin{equation}\label{eq:FTC}
\int_\uu^\vv d\xx f(\xx)=\sum_{J\subseteq[p]}(-1)^{p-|J|}F(\vv_J),
\end{equation}
where
$$\int_\uu^\vv d\xx\; h(\xx):=
(-1)^{\sum_{r=1}^p 1(u_r>v_r) }\int_{[\uu\wedge\vv,\uu\vee\vv]} d\xx\; h(\xx);$$
\begin{equation*}
\uu\wedge\vv:=(u_1\wedge v_1,\dots,
u_p\wedge v_p); \quad
\uu\vee\vv:=(u_1\vee v_1,\dots,u_p\vee v_p);
\end{equation*}
$u\wedge v:=\min(u,v)$ and $u\vee v:=\max(u,v)$ for real $u$ and $v$;
$[\uu,\vv]:=\prod_{r=1}^p[u_r,v_r]$;
$|J|$ is the cardinality of $J$;
$\vv_J:=\uu\1_{[p]\setminus J}+\vv\1_J=\uu+(\vv-\uu)\1_J$;
$\1_J:=(1(1\in J),\dots,1(p\in J))$; $\uu\vv:=(u_1v_1,\dots,u_pv_p)$.
