Marco Grandis has done some work on this, and you can extract answers for 1-3 from these papers
- Finite Sets and Symmetric Simplicial Sets - M Grandis - TAC (pdf)
- Higher Fundamental Functors for Simplicial Sets - M Grandis - CTGDC (pdf)
See also
- An Alternative Presentation of the Symmetric-Simplicial Category - Eric R. Antokoletz - arxiv (link)
Question 1 and 2
The first paper by Grandis above gives a nice detailed overview of all this, including
- a description of (a skeleton of) $\mathbf{FinSet}$ as the walking symmetric strict monoidal category with a commutative monoid
- in terms of generators and relations, with generators faces + degeneracies + transpositions, and relations the standard ones for faces + degeneracies, the Moore ones for transpositions, and some compatibility rels between those.
About who first proved this kind of things, in the second paper, he acknowledges that:
In November 1998, at a PSSL meeting in Trieste, Bill
Lawvere suggested I might extend my study of the homotopy of simplicial
complexes to symmetric simplicial sets, on the basis of his draft [19] where the
fundamental groupoid of the latter is presented as a left adjoint. I would like to
express my gratitude for his kind encouragement and for helpful discussions.
[19] is Lawvere's Toposes generated by codiscrete objects, in combinatorial topology and functional analysis from 1989, which I don't have access to; maybe there's some more info in there.
Question 3
You could find more about this in the second paper by Grandis. The idea is that the classical homotopy theory of simplicial complexes can be extended to symmetric simplicial sets (presheaves on $\mathbf{FinSet}$), so that the edge-path groupoid of a simplicial complex can be identified with what Grandis calls the intrinsic fundamental groupoid, which is the left adjoint of a symmetric nerve (this goes back to Lawvere notes ref above). See also
- An intrinsic homotopy theory for simplicial complexes, with applications to image analysis - M Grandis (pdf)
Thanks to a very helpful discussion with Clark Barwick in the homotopy theory chat, I think I now understand what's going on here. In particular, the ring spectrum structure on the sphere spectrum $\mathbb{S}$ does come from a monoidal structure on $\text{Bord}$, but I was confused about how to transport this monoidal structure from the category to the spectrum.
The monoidal structure can be thought of as coming from the universal property of $\text{Bord}$: since it's the free symmetric monoidal $\infty$-category with duals on a point, the $\infty$-category of symmetric monoidal functors $\text{Bord} \to \text{Bord}$ can canonically be identified with $\text{Bord}$ itself, and hence $\text{Bord}$ naturally acquires a monoidal structure, which I'll call $\circ$, coming from composition of functors $\text{Bord} \to \text{Bord}$. This is exactly analogous to how the free abelian group $\mathbb{Z}$ on a point canonically acquires a ring structure, and compatible under group completion with how the free spectrum on a point, namely $\mathbb{S}$, canonically acquires a ring spectrum structure. Here we need to know that symmetric monoidal $\infty$-categories are enriched over themselves, but this ought to be true by analogy both with the case of abelian groups and with the case of spectra.
This is admittedly an indirect description. It's hard to attempt a more direct description because the resulting monoidal structure isn't all that interesting on objects, and trying to describe what it does on morphisms is what got us into this mess in the first place.
So let me trudge on. In the comments I explained that I thought a monoidal structure
$$\text{Bord} \times \text{Bord} \to \text{Bord}$$
couldn't induce the usual multiplication map on $\mathbb{S}$ because monoidal structures take an $n$-morphism and an $n$-morphism and return another $n$-morphism: for example, the disjoint union does provide a monoidal structure of this form (in fact it's the monoidal structure figuring in the universal property), and the induced map
$$\pi_n(\mathbb{S}) \times \pi_n(\mathbb{S}) \to \pi_n(\mathbb{S})$$
is the usual abelian group structure on $\pi_n(\mathbb{S})$.
The problem with this story as applied to $\circ$ is that, with the natural symmetric monoidal structures on both sides, $\circ : \text{Bord} \times \text{Bord} \to \text{Bord}$ is not a symmetric monoidal functor, so it does not induce a map $\mathbb{S} \times \mathbb{S} \to \mathbb{S}$ of spectra. This issue shows up already at the level of abelian groups: with the natural abelian group structures on both sides, the multiplication map $\mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$ is not a homomorphism of abelian groups.
As suggested by the analogy to abelian groups, $\circ$ is "bilinear": it preserves disjoint unions separately in both variables. So the fix is to use a suitable notion of "tensor product" of symmetric monoidal $\infty$-categories (suitable meaning in particular that on symmetric monoidal $\infty$-groupoids, thought of as connective spectra, it reproduces the smash product) and think of $\circ$ as instead providing a symmetric monoidal functor
$$\text{Bord} \otimes \text{Bord} \to \text{Bord}$$
which reproduces the usual ring spectrum structure $\mathbb{S} \otimes \mathbb{S} \to \mathbb{S}$ (here I am using $\otimes$ for the smash product as well). Now to see how we get a multiplication map
$$\pi_n(\mathbb{S}) \times \pi_m(\mathbb{S}) \to \pi_{n+m}(\mathbb{S})$$
it suffices to recall that the smash product of $S^n$ and $S^m$ is $S^{n+m}$.
So, one way to answer the conceptual question "how, in this situation, did we start with an $n$-morphism and an $m$-morphism and get an $n+m$-morphism" is that the universal property of $\text{Bord}$ is extremely general: it naturally acts by endomorphisms on an object in any symmetric monoidal $\infty$-category with duals whatsoever, including the "loop spaces" of $\text{Bord}$ itself! The analogous statement in stable homotopy is that the stable homotopy groups naturally give rise to operations on the homotopy groups of any spectrum whatsoever, including the shifts of the sphere spectrum itself.
Best Answer
Infinite loop spaces and spectra are intrinsically pointed, and the purpose of the basepoint is to build in basepoints, which give the units for the associated products. Let $T_*$ be the category of based objects in any cartesian monoidal category T. For an object $X$ of $T_*$, a covariant functor $X^*: F_* \longrightarrow T_*$ that sends the based set $n=\{0,1, \cdots, n\}$ with basepoint $0$ to $X^n$ is precisely a commutative monoid in $T$ with its unit element equal to the basepoint of $X$. Use of basepoints like this long precedes Segal. When $T$ is spaces, the James construction $JX$ is the free monoid on $X$ with unit the basepoint of $X$ and the infinite symmetric product $NX$ on $X$ is the free commutative monoid on $X$ with its unit element the basepoint of $X$, both suitably topologized.
In more detail, the morphisms of $F$ are generated under composition by injections, projections, and the based maps $\phi_n \colon n\to 1$ that send $i$ to $1$ for $1\leq i\leq n$. (Using the wedge sum, only $\phi_2$ need be added). The morphisms $\pi\colon m\to n$ such that $\phi^{-1}(i)$ has $0$ or $1$ element give a subcategory $\Pi$ of $F$, and the functor $X^*$ has underlying functor $\Pi\to T_*$ given by the injections (determined by the basepoint), projections, and permutations that are given by the assumption than $n \mapsto X^n$. The map $\phi_2$ gives a product $X\times X\longrightarrow X$ and, more generally, the $\phi_n \colon X^n\longrightarrow X$ give the unique $n$-fold product determined by $\phi_2$.
The point of infinite loop space theory is to build in the axioms of a commutative monoid up to "all higher coherence homotopies", and the genius of Segal was to see that the evident maps $\delta_i\colon n\longrightarrow 1$ that send $i$ to $1$ and $j$ to $0$ for $j\neq i$ can be used to build in these homotopies. Taking $T$ to be spaces, for a functor $Y\colon F_* \longrightarrow T_*$, we have based spaces $Y_n$, and the $\delta_i$ determine the Segal maps $\delta^n\colon Y_n \longrightarrow Y_1^n$. Requiring these maps to be homotopy equivalences for all $n\geq 0$ makes $Y_1$ a "commutative monoid up to all higher coherence homotopies", with canonical zigzag product $Y_1^2 \longleftarrow Y_2 \longrightarrow Y_1$ determined by $\delta^2$ and $\phi_2$.
I could go on forever about this. But maybe I'll just give the definition of the functor $K\colon \Delta^{op}\longrightarrow F_*$. I agree that its interpretation may not be obvious. Of course, $Kn = n$. For a map $\phi\colon n\longrightarrow m$ in $\Delta$, define $K\phi\colon m\longrightarrow n$ by sending $i$ to $j$ whenever $\phi(j-1) < i \leq \phi(j)$, where $1\leq j\leq n$, and sending $i$ to $0$ if there is no such $j$. Thus $$ (K\phi)^{-1}(j) = \{ i | \phi(j-1) < i \leq \phi(j)\} \ \ \text{for} \ \ 1\leq j\leq n. $$
Please excuse the lousy ad hoc notation (F= finite sets).