Multisections of the universal curve

Question

Fix some $g \geq 2$. Let $\mathcal{M}_g$ be the moduli space of smooth genus $g$ curves over $\mathbb{C}$. For some $d \geq 1$, let $X_{g,d} \rightarrow \mathcal{M}_g$ be the family whose fiber over $S \in \mathcal{M}_g$ is the $d^{\text{th}}$ symmetric power of $S$. I'm aware that due to the presence of curves with automorphisms this doesn't (strictly speaking) exist, but let's ignore that point (e.g. by adding a full level structure to $\mathcal{M}_g$ to rigidify things).

Question 1: Does there exist some $d \geq 1$ such that $X_{g,d} \rightarrow \mathcal{M}_g$ has a section? I expect something like Weierstrass points will work here, but I don't know how they vary in families.

Question 2: Assuming that Question 1 has a positive answer, what I'm really interested in is the following. Does there exit some $d,e \geq 1$ such that there exist sections $\sigma\colon \mathcal{M}_g \rightarrow X_{g,d}$ and $\sigma'\colon \mathcal{M}_g \rightarrow X_{g,e}$ such that for all $S \in \mathcal{M}_g$, the $d$ points making up $\sigma(S)$ are disjoint from the $e$ points making up $\sigma'(S)$? Here I don't have a candidate for the two disjoint multisections.

Answer 1

I would like to give an answer to the simple Question 1. The answer relies on the following lemma.

Lemma. For any fixed $(r,g,n)$ there exists a hypersurface $V\subset \mathbb CP^n$ that doesn't contain any smooth genus $g$ curve of degree $\le r$.

I think, this lemma can be proven by simple dimension count.

Now, to solve the question, find $n$, such that (the universal curve) $M_{g,1}$ can be embedded in $\mathbb CP^n$. Let $r$ be the degree of all the genus $g$ curves in the embedding. Take, a hypersurface $V$ from the Lemma and let $d$ be its degree. Now, take and the intersection of $V$ with each curve. This will give you the desired section.

(As for Question 2, I have no clue).