About a generalization of the Borsuk-Ulam theorem I've come upon this MO post, and I was wondering if there is a way to generalize what is said in the answer at the very bottom. Specifically, we have that given continuous maps $f: \mathbb{S}^n \to \mathbb{S}^n$, and $g: \mathbb{S}^n \to \mathbb{R}^n$, if $f$ is an involution, then there is a point $x \in \mathbb{S}^n$ such that $g(x) = g(f(x))$, i.e. the value at $x$ is fixed under the transformation of the sphere by $f$.
We know from the answer of the post that for a general $f$, there need not be such a fixed point. But now what if instead $f$ is such that there exists a $k \in \mathbb{N}$ where $f^k$ is the identity map, do we still get such a fixed point? Furthermore, do we know about other spaces besides the sphere that have this property when $f$ is an involution?
 A: A first observation is that a map $f:S^n\to S^n$ with $f^k=\operatorname{Id}_{S^n}$ generates a topological action of the cyclic group $\mathbb{Z}/k$ on $S^n$. So we are talking about generalising from $\mathbb{Z}/2$ actions to $\mathbb{Z}/k$ actions.
There are many generalisations of the Borsuk-Ulam theorem of this kind in the literature. Most of them concern estimates of the dimension of the set
$$
A(g)=\{x\in S^n \mid g(x)=g(f^i(x))\mbox{ for }i=1,\ldots , k\},
$$ for various flavours of map $g:S^n\to M^m$. 
Interpreted this way, the answer to your first question appears to be negative in general, by a theorem of Munkholm in 
Munkholm, H. J., On the Borsuk-Ulam theorem for (Z_{p^ a}) actions on (S^{2n-1}) and maps (S^{2n-1} \to R^ m), Osaka J. Math. 7, 451-456 (1970). ZBL0211.55701.
In particular, let $k=p^a\neq 9$ be an odd prime power with $a>1$, and let $\omega=\operatorname{exp}(2\pi i/p^a)$ be a primitive $p^a$-th root of unity. Let $f:S^{2n-1}\to S^{2n-1}$ be given in complex coordinates by
$$
f(z_1,\ldots , z_n)=(\omega z_1,\ldots ,\omega z_n).
$$
Then Munkholm's "mod $p^a$ Borsuk-Ulam anti-theorem" seems to imply that there exists a continuous map $g:S^{2n-1}\to \mathbb{R}^{2n-1}$ with $A(g)=\emptyset$.
For a summary of positive results in the literature, you could do worse than Chapter 1 of Yuri Turygin's thesis at the Unviersity of Florida. 
