First, here are some things about the four generalizations you mention:
Monoids don't fall into Diers' framework: By his Proposition 1.4.1 the terminal object in his framework is strict, i.e. any morphism $1 \to A$ is an isomorphism, which is definitely not the case for monoids.
I also wouldn't expect Diers' examples to be instances of Toen/Vaquie's framework in general, Diers' example 1.3.16, the category of pairs (ring, module over it), might be a counterexample. I don't know about Durov's setting.
Durov's geometry is in no obvious way an instance of Toen/Vaquie's framework. If you want to force it into that framework, this might be an idea to go after: Monads are monoids in the monoidal category of endofunctors. Commutative monads, however, are not commutative monoids in that category; indeed it doesn't even make sense to say that since the category is not symmetric monoidal. So first you have to find a symmetric monoidal ambient category in which Durov's generalized rings live. Seeing monads as Lawvere algebraic theories or as (things presented by) sketches might do the job - a commutative theory is probably exactly a sketch with an isomorphism from its tensor square. Another idea could be to consider a category of monads where morphisms are natural monad transformations together with distributive laws...
Derived algebraic geometry on the other hand is an instance of the homotopical version of Toen/Vaquie's framework, also contained in that article - see also below.
Second let me point out that there are many more generalizations of algebraic geometry than those four:
° Rings with extra structure can count as generalization, if one can endow any usual ring with such an extra structure, e.g.
Not Borger's geometry with lambda-rings: Not any ring can be endowed with the trivial lambda-ring structure - see his comment
Berkovich's analytic geometry: Any ring can be endowed with the trivial metric
° One can replace rings by first order structures in several ways:
Several Russian authors do this in somewhat similar ways, a recent reference is this one by Daniyarova, Myasnikov, Remeslennikov which has many references to other work in this direction; see also this one by B. Plotkin.
First order structures can be described by sketches and there is an outline of algebraic geometry along this line in this text by R. Guitart.
° There are hyperrings (used for algebraic geometry by Connes/Consani) and fuzzy rings (by Walter Wenzel and Andreas Dress, e.g. this), which are certain second order structures.
° There are two generalizations of rings used by Shai Haran to compactify the integers, F-rings and the one given in his "Non-additive prolegomena".
° There is another generalization made by Shai Haran in his article on "dyslectic geometry" in which rings are endowed with gradings over general monoids (see here).
Something quite similar seems to be going on in James Dolan's generalization of algebraic geometry (unpublished, but see here, there also was a series of videos of talks somewhere)
° derived algebraic geometry doesn't necessarily have to be based on simplicial rings; dg-algebras and $E_\infty$-ring spectra are equally important inputs, and there are many others, captured in
Toen-Vezzosi's HAG-contexts (these are homotopically additive)
Lurie's structured spaces from DAG 5 which capture about everything based on the idea of glueing together homotopical algebraic structure.
° note that Toen/Vaquie in their relative geometry do not stop at commutative monoids in some monoidal category but also give a homotopical version - this is something like a non-additive version of HAG-contexts and covers e.g. geometry over the "spectrum with one element" whose input are simplicial monoids.
° replacing rings by groupoid objects in rings together with an appropriate notion of equivalence gives you stack theory. Of course you can go on to higher stacks.
° of course there are the several approaches to non-commutative algebraic geometry - see Mahanta's survey for some of them, many related to the next point
° Chirvasitu/Johnson-Freyd's 2-schemes
° Takagi's generalized schemes
° Deitmar's congruence schemes
° Lorscheid's blue schemes
° I am sure I forgot several things...
Third, since I am at it, let me note that there are also generalizations of algebraic geometry which do not exactly build upon a generalized notion of ring, e.g.
° Hrushovski/Zilber's Zariski Geometries (see here) capturing the essential structure which is used in the applications of model theory to arithmetic geometry
° Rosenberg's noncommutative geometry. It doesn't have to be non-commutative, and also not additive, as Z. Skoda pointed out here
° in particular: schemes as dg-categories (Kontsevich, Rosenberg, Tabuada)
° Balmer's triangulated geometry
...
To summarize: Carefully mapping AG-land will keep you busy for a while. I have gathered quite some material for a rudimentary map (or maybe a low resolution satellite photo) accompanied by a few selected closer snapshots, but I won't start writing it before the second half of the year...
$\newcommand{\N}{\mathbb N}\newcommand{\paren}[1]{\left(#1\right)}\newcommand{\T}{\mathbb{T}}\newcommand{\m}{\mathfrak{m}}\newcommand{\E}{\mathbf{E}}$I can answer your first set of questions:
There is a geometric theory of blueprints. It's easiest* to work from the definition of a blueprint as a pair $B = (A,R)$ consisting of a semiring $R$ and a multiplicative subset $A \subseteq R$ which contains $0$ and $1$, and which generates all of $R$.
The theory will have a sorts $A,R$, function symbols $+, \cdot$, constant symbols $0_A,1_A, 0_R, 1_R$, and a unary function symbol $\iota$, giving the inclusion of $A$ into $R$. In addition to axioms asserting that $R$ is a semiring and that $A$ embeds monomorphically as a multiplicative submonoid of $R$ such that $\iota(0_A) = 0_R$ and $\iota(1_A) = 1_R$, we have the infinitary axiom
$$ \vdash^{x : R} (x = 0) \lor \paren{\bigvee_{n \in \N, \; \varphi \in \text{Oper}_{n+1}}\exists a_0\dotsm a_n . \varphi(\iota(a_0),\dotsc,\iota(a_n)) = x }$$
where $\text{Oper}_n$ is the set of $n$-ary semiring operations built from $0_R,1_R,+,\cdot$. This axioms states that $A$ suffices to generate all of $R$
There can be no coherent axiomatization of the theory $\T$ of blueprints. To see this, suppose that $\T$ were coherent. Then we could obtain a new coherent theory $\T\,'$ by introducing the following additional coherent axioms which require a blueprint in $\text{Set}$ to be isomorphic to $(\{0,1\} \hookrightarrow \N)$.
$$
\vdash^{a: A} \iota(a) = 0 \lor \iota(a) = 1
$$ $$
x + y = 0 \vdash^{x,y : R} x = y = 0 $$
Since any consistent finitary first-order theory with an infinite model will admit arbitrarily large models in $\text{Set}$, this is impossible.
Regarding local blueprints: The definition of local blueprints as those having a unique maximal ideal of course cannot, in its current form, be stated in geometric logic. However, we can say in $\text{Set}$ that a congruence $\sim_\m$ on a blueprint $B = (A,R)$ is the unique maximal nontrivial congruence on $B$ iff for any pair of elements $x,y \in R$, if $x \sim_\m y$ fails, then the smallest congruence $\sim$ such that $x \sim y$ is trivial. This can be stated in a conservative geometric extension of our theory of blueprints if we adjoin a binary relation symbol $\sim_m$ on $R \times R$, together with axioms stating that $\sim_m$ is a congruence, in addition to the following axiom which states that unless $x \sim_\m y$ holds, every congruence containing $(x,y)$ contains every pair of elements in $R$.
$$\vdash^{x,y,z,w:R} {x \sim_\m y \lor} \paren{\bigvee_{n \in \N} \exists a_0\dotsm a_n : A . \E(z,\iota(a_0)) \land \E(\iota(a_0), \iota(a_1)) \land \dotsi \land \E(\iota(a_n), w)}
$$ where $\E(c,d)$ denotes the sub-expression $$
\bigvee_{n \in \N, \; \varphi \in \text{Oper}_{n+2}} \exists b_0 \dotsm b_n : A. \varphi(c,b_0,\dotsc,b_n) = \varphi(d,b_0, \dotsc, b_n)
$$
*But not essential. Since the list object is a geometric construction, we could also write down a two-sorted theory which axiomatizes the behavior of an equivalence relation on the set of lists. The downside of that approach is that it would involve lots of complicated-to-state infinitary axioms.
Best Answer
This "answer" will basically restate the comments of Marty and Jason Starr.
Any variety covered by schemes of the form $\mathrm{Spec}(\mathbf Z[M_i])$, or any torified variety, is rational. And indeed Severi conjectured at one point that $M_g$ is rational for any $g$! But we know a lot about the Kodaira dimension of $M_g$ by work of Harris--Mumford, Farkas, Eisenbud, Verra, ... in particular we know that Severi's conjecture is maximally false. We have $\kappa(M_g) = -\infty$ for $g \leq 16$, we don't know anything for $17 \leq g \leq 21$, and $\kappa(M_g) \geq 0$ for $g \geq 22$. In fact we know that $M_{g}$ is of general type for $g = 22$ and $g \geq 24$. But if one only wants an example of a $g$ for which $M_g$ is not rational, then I think one can find examples much earlier (maybe $g \approx 6,7$?).
See http://arxiv.org/abs/0810.0702 for a survey by Farkas.
Moreover, the cohomology of a torified scheme can only contain mixed Tate motives. In particular, this implies properties like: the number of $\mathbf F_q$-points is a polynomial function of $q$. As Marty says there are no problems in genus zero; these might be honest-to-God $\mathbf F_1$ schemes. The cohomology of $M_{1,n}$ contains only mixed Tate motives when $n \leq 10$, but for $n \geq 11$ one finds motives associated to cusp forms for $\mathrm{SL}(2,\mathbf Z)$. (The number 11 arises as one less than the smallest weight of a nonzero cusp form; the discriminant form $\Delta$.) This implies that the polynomiality behaviour changes drastically -- the number of $\mathbf F_q$-points is now given by Fourier coefficients of modular forms, which are far more complicated and contain lots of arithmetic information.
The connection with birational geometry is also very visible here. The cohomology classes on $M_{1,n}$ associated to the cusp forms of weight $n+1$ are of type $(n,0)$ and $(0,n)$ in the Hodge realization, since given such a cusp form one can explicitly write down a corresponding differential form in coordinates. So by the very definition of Kodaira dimension we can not have $\kappa = -\infty$ anymore.
Bergström computed the Euler characteristic of $M_{2,n}$ in the Grothendieck group of $\ell$-adic Galois representations by point counting techniques. That is, he found formulas for the number of $\mathbf F_q$-points of $M_{2,n}$ by working really explicitly with normal forms for hyperelliptic curves. These formulas turned out to be polynomial in $q$ for $n \leq 7$ (and conjecturally for $n \leq 9$), just as they would be if we knew that the cohomology of $M_{2,n}$ contained only mixed Tate motives. By thinking a bit about the stratification of topological type, one concludes that this holds also for $\overline M_{2,n}$ when $n \leq 7$, which is smooth and proper over the integers. Then a theorem of van den Bogaart and Edixhoven implies that the cohomology of $\overline M_{2,n}$ is all of Tate type and with Betti numbers given by the coefficients of the polynomials.
There are similar results by Bergström for $M_{3,n}$ when $n \leq 5$ and Bergström--Tommasi for $M_4$. But the general phenomenon is that increasing either $g$ or $n$ will rapidly take you out of the world of $\mathbf F_1$-schemes, at least if this is taken to mean "commutative monoids".
However, I don't know enough $\mathbf F_1$--geometry to say what the answer is if one takes $\mathbf F_1$-schemes in the sense of Borger. The first nontrivial case to answer would be: are the motives associated to cusp forms for $\mathrm{SL}(2,\mathbf Z)$ defined over $\mathbf F_1$ in his set-up? To be clear, I don't believe that this is true, but I know almost nothing about $\lambda$-schemes.
Let me also make two small remarks: (i) Everything I have written above is necessary conditions for being defined over $\mathbf F_1$. (ii) The fact that $M_{g,n}$ is smooth over the integers says that the cohomology of $M_{g,n}$ is at least quite "special", even though it is not defined over $\mathbf F_1$. Smoothness is a strong restriction on the Galois representations that can occur in the cohomology.