If you are happy to restrict attention to the interior, then in a two-good-two-agent economy - given differentiable, strictly concave, and increasing utility functions - the Pareto curve must be increasing. Then, $\textit{a fortiori}$, the contract curve is also increasing.
To see the Pareto set is representable by an increasing function in this case, first recall that given differentiability, strict concavity, and monotonicity, an interior allocation is Paretian if and only if $MRS_1(x_{11},x_{12})=MRS_2(x_{21},x_{22})$. Therefore, given an economy endowment of $(\omega_1,\omega_2)$, the Pareto set is implicitly defined by the following relation (preferences are locally non-satiated so the endowment constraints bind), $$P(x_{11},x_{12}):=MRS_1(x_{11},x_{12})-MRS_2(\omega_1-x_{11},\omega_2-x_{12})=0$$
Applying the implicit function theorem, we obtain $$\frac{d x_{12}}{d x_{11}}=-\frac{\partial_{x_{11}}P}{\partial_{x_{12}}P}=-\frac{\partial_{x_{11}}MRS_1+\partial_{x_{21}}MRS_2}{\partial_{x_{12}}MRS_1+\partial_{x_{22}}MRS_2}$$
Given strict concavity (which implies decreasing $MRS$), we have $$\begin{align}\partial_{x_{11}}MRS_1<0 \\ \partial_{x_{21}}MRS_2<0 \\ \partial_{x_{12}}MRS_1>0 \\ \partial_{x_{22}}MRS_2>0 \end{align}$$ Therefore, over the Pareto set $$\frac{d x_{12}}{d x_{11}}>0$$ Whence, as the contract curve is a subset of the Pareto set, it is also an increasing curve in the Edgeworth box. Additionally, since the curve is strictly increasing, this also proves the Pareto set relation cannot be many-to-one.
As for your question about whether the Pareto set relation can be one-to-many, the implicit function theorem proves the existence of a unique $\textbf{function}$ (so not a one-to-many relation) characterising the Pareto set. More explicitly, decreasing $MRS$ removes one-to-many behaviour as a possibility. Suppose there is some $(\tilde{x}_{11},\tilde{x}_{12})$ which is Paretian. Fix some $\hat{x}_{12}> \tilde{x}_{12}$ which is feasible. By decreasing $MRS$, $$ \begin{align} MRS_1(\tilde{x}_{11},\hat{x}_{12})&>MRS_1(\tilde{x}_{11},\tilde{x}_{12}) \\
MRS_2(\omega_1-\tilde{x}_{11},\omega_2-\hat{x}_{12})&<MRS_2(\omega_1-\tilde{x}_{11},\omega_2-\hat{x}_{12})
\end{align}$$ So then $P(\tilde{x}_{11},\hat{x}_{12})>P(\tilde{x}_{11},\tilde{x}_{12})=0$ meaning $(\tilde{x}_{11},\hat{x}_{12})$ cannot be Paretian as $MRS_1>MRS_2$. An analogous argument works, $\textit{mutatis mutandis}$, to show no $\hat{x}_{12}<\tilde{x}_{12}$ can result in a Pareto allocation either. Hence, the Pareto set correspondence cannot be one-to-many.
