The question seems to be about algebraic geometry of commutative semirings (these are rings without subtraction).

The theory by Toen-Vaquié (and others) in "Au-dessous de $Spec \mathbb{Z}$" develops (functorial) algebraic geometry relative to any bicomplete closed symmetric monoidal category. The usual case is $\mathcal{V}=(\mathsf{Ab},\otimes)$, since then $\mathsf{CommMon}(\mathcal{V}) \cong \mathsf{CommRing}$. Now take $\mathcal{V}=(\mathsf{CommMon},\otimes)$ (the tensor product is defined and constructed in the same fashion as the one of abelian groups), then $\mathsf{CommMon}(\mathcal{V})\cong \mathsf{CommSemiRing}$. The category of schemes relative to $(\mathsf{CommMon},\otimes)$ is called the category of $\mathbb{N}$-schemes by Toen-Vaquié (since $(\mathbb{N},+,\cdot)$ is the initial commutative semiring). They also compare this category to the category of $\mathbb{F}_1$-schemes (which corresponds to $\mathcal{V}=(\mathsf{Set},\times)$, or rather $\mathcal{V}=(\mathsf{Set}_*,\wedge)$).

Alternatively, we may view $\mathbb{N}$ as a generalized commutative ring (commutative algebraic monad on $\mathsf{Set}$) and hence use Durov's notion of a generalized scheme ("New Approach to Arakelov Geometry"), which has a more geometric flavor; these are locally ringed spaces which are locally isomorphic to affine schemes, which in turn consist of prime ideals and a structure sheaf defined by localizations. Thus, the construction of the category of $\mathbb{N}$-schemes is very similar to the construction of the category of $\mathbb{Z}$-schemes, except that we ignore the non-existing subtraction.

A commutative semiring is simple (i.e. has exactly two quotients) if and only if it is non-trivial and every non-zero element is invertible with respect to multiplication; such objects are sometimes called semifields. Thus, $\mathbb{R}_{\geq 0}$ is an example of a semifield.

I don't know how much has been done on étale morphisms and Galois theory in relative algebraic geometry, specifically for $\mathbb{N}$-schemes (hopefully, others can write something about this), but the notion of smooth morphisms is the main topic in Florian Marty's thesis "Des Ouverts Zariski et des morphisms formalement lisse en géométrie relative", and in the introduction he hints at a definition of étale morphisms as smooth morphisms with a smooth diagonal.

Here is another idea (which should work at least for $\mathbb{N}$-schemes): The étale morphisms are precisely the flat unramified morphisms. The notion of flatness is defined as usual for affine relative schemes and then by glueing to arbitrary relative schemes. Unramifiedness may be characterized as being locally of finite presentation, which has a common functorial characterization, and the vanishing of the quasi-coherent sheaf of differentials, which may be constructed as usual via glueing.

I don't know anything specifically about the étale $\mathbb{R}_{\geq 0}$-schemes, though. My hope is that my "answer" bumps your question and invites others go give proper answers, which I would enjoy to read, too.

**Added**: $\Omega^1_{\mathbb{R}/\mathbb{R}_{\geq 0}}$ is indeed trivial. Since we have the commutative semiring presentation$$\mathbb{R} \cong \mathbb{R}_{\geq 0}[X]/\langle X^2=1,X+1=0\rangle,$$we obtain the $\mathbb{R}$-semimodule presentation
$$\Omega^1_{\mathbb{R}/\mathbb{R}_{\geq 0}} \cong \mathbb{R} \cdot d(X) / \langle X \cdot d(X)=0,d(X)+0=0 \rangle = 0.$$
However, $\mathbb{R}_{\geq 0} \to \mathbb{R}$ is not flat: Notice that $\mathbb{R}$ is the $\mathbb{R}_{\geq 0}$-semimodule freely generated by $1$ and $-1$ modulo the relation $1+(-1)=0$. Thus, if $M$ is some $\mathbb{R}_{\geq 0}$-semimodule, then $M \otimes_{\mathbb{R}_{\geq 0}} \mathbb{R}$ is the quotient of $M \oplus M$ by the congruence generated by $\{(m,m) : m \in M\}$. This is exactly the Grothendieck group $G(M)$ with the induced $\mathbb{R}$-action. Now if $M \to N$ is an injective homomorphism of $\mathbb{R}_{\geq 0}$-semimodules, it does not follow that $G(M) \to G(N)$ is an injective homomorphism of $\mathbb{R}$-modules. For example, let $M=\mathbb{R}_{\geq 0}$ and $N$ be the $\mathbb{R}_{\geq 0}$-semimodule freely generated by two elements $x,y$ with $x+y=y$. Then $M \to N$, $1 \mapsto x$ is injective, but $G(M) \to G(N)$ vanishes.