Let me answer the OP's side question:

"As a side note, I also realized that I do not know any ring whose field of fractions is $\mathbb{R}$."

Sure you do: $\mathbb{R}$! But I know what you mean: you are asking whether there is a **proper** subring $R$ of $\mathbb{R}$ such that $\mathbb{R}$ is the field of fractions of $R$. The answer is **yes**. (Among the choices you provide, I would lean towards the second. The proof that follows certainly uses AC. Whether it *must* use AC is not really my bag, but I would guess that it does.) This follows from the following stronger result.

**Theorem**: For a field $K$, the following are equivalent:

(i) $K$ admits a nontrivial $\mathbb{R}$-valued valuation.

(ii) $K$ is *not* algebraic over a finite field.

This result is proved in these course notes of mine. Well, at least morally: at the moment I can track down a Remark on p. 6 that it *will* be proved and the result which is the essential content, Theorem 30b) in $\S$ 1.3:

**Extension Theorem** If $K$ is a field and $|\cdot|$ is a nontrivial non-Archimedean norm on $K$ -- i.e., such that taking $-\log |\cdot|$ gives a nontrivial $\mathbb{R}$-valued (or "rank one") valuation -- then for *any* extension field $L/K$, the norm $|\cdot|$ extends to $L$.

From this the above proof follows easily: for (i) $\implies$ (ii) just notice that for every element $x$ of finite order in the multiplicative group of a field $K$ we must have $|x| = 1$ for every norm $|\cdot|$. A field $K$ which is algebraic over a finite field has $K^{\times}$ a torsion group: every element has finite order. For (ii) $\implies$ (i) notice that every field $K$ which is not algebraic over a finite field can be expressed as an extension field of either $\mathbb{Q}$ or $\mathbb{F}_p(t)$, and each of these fields famously carries plenty of nontrivial non-Archimedean norms. Apply the Extension Theorem.

To get an answer to your question: let $|\cdot|$ be a nontrivial non-Archimedean norm on $\mathbb{R}$ and let $R$ be its valuation ring. This is indeed a domain, proper in $\mathbb{R}$, with fraction field $\mathbb{R}$.

Now finally one might ask: okay, such rings exist, but who cares? Surely this is just a curiosity. To that I would say: you *must* read this four page note of Paul Monsky, published in the MONTHLY in 1970. And you had better hold on to your hat while you read it, and give some attention to your socks as well: they are in grave danger of being knocked off. Monsky's Theorem was introduced to me by Aaron Abrams in 2006. It was in fact much of the motivation for the material on extension of valuations which is included in the notes I linked to (although, sadly, I have not gotten around to writing up the application to Monsky's Theorem).