If $\mathcal{C}$ is a stable $\infty$-category, then the endomorphisms of $\iota_{\mathcal{C}}$ has the structure of an $E_2$-ring spectrum (the "Hochschild cohomology" of $\mathcal{C}$). Making $\mathcal{C}$ $R$-linear is equivalent to giving an $E_2$-map from $R$ into this endomorphism ring. And $\mathbf{F}_p$ has a very simple presentation as an $E_2$-ring spectrum: you need only the single relation ``$p=0$'' (result of Mahowald when $p=2$, Hopkins for odd primes).

The localization can be described more concretely: it's sheaves with respect to the Grothendieck topology where every morphism generates a covering. Equivalently, it's functors F: {finite spaces} -> {spaces} (or you can do a pointed version if you like) with the property that for any X, F(X) is the totalization of the cosimplicial space given by applying F to the "Cech nerve" of the map X->* (in the opposite of finite spaces). This is much larger than the class of 1-excisive functors: it contains all n-excisive functors for any n, and more (such as products of n-excisive functors as n varies).

True iff the unit for the monoidal structure is a compact generator.

Not sure if it is explicitly stated there, but it is at least easily derived from what's there. First, reduced 1-excisive functors are spectra (Calculus III). If F is an arbitrary excisive functor then Y=F(*) is a space and $(y \in Y) \mapsto \lambda X. F(X) \times_{F(*)} \{y\}$ is a local system of reduced excisive functors on $Y$. Conversely if $\{ F_y \}_{y \in Y}$ is a local system of reduced excisive functors on a space Y, then $F(X)=\varinjlim_{y \in Y} F_y(X)$ is an excisive functor (not necessarily reduced). These constructions are homotopy inverse to one another.

Tom Goodwillie, "Calculus III: Taylor Series"

I'm afraid that I don't follow either. What exactly are you saying you can rule out?

@Ulrik That paper looks quite relevant, but not exactly what I'm looking for. If I'm reading it right, it seems to be about functions whose totality is provable using very weak set-theoretic assumptions, analogous to the characterization of primitive recursive functions as those functions which are provably total using very weak arithmetic assumptions. But I'm hoping for something which is specific to a fixed admissible set $A$, and specializes to primitive recursive arithmetic when I take $A = HF$. (Something that would generalize PRA, rather than being analogous to PRA.)

@Carlo It is relevant, but I am asking for something more refined: I want to consider not just which sets are $\Sigma_1$-definable in $A$, but when $A$ "knows" that one $\Sigma_1$-set is contained in another.