Is there a subfield of $\mathbb{R}$ that admits a non-Archimedean order? We know any Archimedean ordered field can be embedded as a subfield of $\mathbb{R}$, but is there a subfield, on which there is another order instead of the one imposed by $\mathbb{R}$, such that it’s not Archimedean?

Yes. For instance, if $\alpha\in\mathbb{R}$ is any transcendental number, then you can order the field $\mathbb{Q}(\alpha)\subset\mathbb{R}$ such that $\alpha$ is infinitely large. Explicitly, you can say a rational function in $\alpha$ is positive iff the leading coefficients of its numerator and denominator have the same sign.