$n$-point map $S$ from $k$-manifold $X$ to $\mathbb{R}^k$

It is known that for every continuous map $$M:S^1\to \mathbb{R}$$ there are infinitely many double points. Also by the Borsuk-Ulam theorem, this is true for each continuous map $$N:S^n\to \mathbb{R}^n$$, $$n\in \mathbb{N}$$. A question arising here concerns finding a (connected, metrizable) $$k$$-manifold $$X$$ for which each continuous map $$S:X \to \mathbb{R}^k$$ has a $$n$$-point, $$2.

A: Does there exist such $$X$$? and if yes then how it would be figured out?

B: Does there exist a minimal $$X$$? (means the number of $$n$$-points of a map $$S:X \to \mathbb{R}^k$$ be minimum among all $$S$$ and all $$X$$.)

Note: a function $$f:X\to Y$$ has a $$n$$-point if for some point $$p\in Y$$ the preimage $$f^{-1}(\{p\})$$ consists of at least $$n$$ -points in $$X$$; a $$2$$-point is also called a double point.