I think the questions were about unconditional proofs or counter examples. I don't have an answer to any of those questions but I think it is still interesting to understand how the yoga of motives suggests natural answers to theses questions. Even though this may seem trivial to people familiar with the subject.

Let's work in the setting of Voevodsky's $\otimes$-triangulated categories $DM^{eff}(\mathbb{Q}):= DM_{gm}^{eff}(Spec(\mathbb{Q});\mathbb{Q}) \subset DM_{gm}(Spec(\mathbb{Q});\mathbb{Q}) =: DM(\mathbb{Q})$. Remember that the latter is obtained by formally inverting $\mathbb{Q}(1)$ and that it is a rigid $\otimes$-triangulated category.

**Question 1**: Is the ring of (effective) periods a field?

Following Beilinson's "Remarks on Grothendieck's standard conjecctures", let's assume

**Motivic conjecture:** There exists a non degenerate t-structure on Voevodsky's category $DM(\mathbb{Q}) := DM(Spec(\mathbb{Q});\mathbb{Q})$ and such that the Betti realization function $\omega_B: DM(\mathbb{Q}) \to D^bMod_f(\mathbb{Q})$ is a $t$-exact $\otimes$-functor.

This is an extremely strong conjecture as it implies the standard conjectures in characteristic 0.

Under this conjecture, the heart of the motvitic $t$-structure is a tannakian category $MM(\mathbb{Q})$. We have Betti and Rham realization functors $\omega_B,\omega_{dR}: MM(\mathbb{Q}) \rightrightarrows Mod_f(\mathbb{Q})$. And we can define
$$
Per := Isom^\otimes(\omega_{dR},\omega_{B})
$$
This is a fpqc-torsor under the motivic Galois group $G_B := Aut^\otimes(\omega_B)$.

Define the algebra of motivic periods as the ring of regular functions on the Betti/de Rham torsor:
$$
P_{mot} := \mathcal{O}(Per)
$$
Integration of differential forms (or more generally the Riemann-Hilbert correspondance) defines an $\mathbb{C}$-point
$$
Spec(\mathbb{C}) \longrightarrow Per
$$
The image of the corresponding morphism $P_{mot} \to \mathbb{C}$ is the ring of periods $P$.

Note: I think this whole part is actually known unconditionnally in the setting of Nori's motives (see arXiv:1105.0865v4).

**Period conjecture:** The morphism $P_{mot} \to P$ is an isomorphism.

Now based on these tiny little conjectures we can say

**Prop:** $P_{mot}$ is not a field so $P$ isn't either.

Proof: Indeed in his comment G-torsor whose ring of regular functions is a field. @quasi-coherent explained how faithfull flatness would imply that if $P_{mot}$ were a field then it would be algebraic over $\mathbb{Q}$ which contradicts the fact that $2\pi i$ belongs to the image of $P_{mot}\to \mathbb{C}$.

**Question 2** Is it true that $(P_{mot}^{eff})^\times = \overline{\mathbb{Q}}^\times$?

This post is getting too long already so I'll try and write down the rest later but the basic idea is that invertible effective motives are Artin motives. This can be proved in terms of weights or niveau (level).