One possible answer to Q1 is theorem 2.7 in this paper, also see the presentation on YouTube by one of the authors.
Roughly, the theorem says that if a quantum field theory
- is of spacetime dimension at least three;
- satisfies relativistic invariance;
- has a compact internal symmetry group;
- the internal symmetry group stays the same when Wick rotating from Lorentzian to Euclidean signature,
then 'spacetime and internal symmetries must be split'.
This brings us also to Q2: as you say the internal symmetry group and the Lorentz group can indeed combine in some minor way, so we must be careful what we mean by 'split' here. For example, in a system without time-reversing symmetries but with fermions, the (double cover of the Euclidean) Lorentz group often combines with $G_{int}$ to give us
$$\frac{\operatorname{Spin}(d) \times G_{int}}{\mathbb{Z}_2^F}.$$
Here we quotient by the diagonal $\mathbb{Z}_2$-subgroup generated by $(-1,(-1)^F)$, where $-1 \in \operatorname{Spin}(d)$ is the nontrivial element in the kernel of the double cover to $SO(d)$ and $(-1)^F \in G_{int}$ is fermion parity.
The situation including time-reversing symmetries is slightly more subtle. I would formulate their formal mathematical definition of 'split' in a language a bit closer to physics as follows. Consider a fermionic symmetry group $G$, which I define to be a compact Lie group $G$ equipped with a homomorphism $\theta: G \to \mathbb{Z}_2 = \{\pm 1\}$ (time-preserving vs time-reversing) and a time-preserving central element $(-1)^F \in G$ of square one.
The prototypical example of such a group is $\operatorname{Pin}^+(d)$, where $(-1)^F$ is the nontrivial element of the kernel of the map to $O(d)$ and $\theta_{\operatorname{Pin}}$ is the composition $\operatorname{Pin}^+(d) \to O(d) \xrightarrow{\operatorname{det}} \mathbb{Z}_2$. Every fermionic symmetry group has a canonical automorphism of order at most two:
$$
g \mapsto
\begin{cases}
g \quad &\theta(g) = 1,
\\
(-1)^F g \quad &\theta(g) = -1.
\end{cases}
$$
Now according to Freed-Hopkins the most general structure group possible we can endow our Euclidean signature quantum field theories with is
$$
\ker\left( \theta: \frac{\operatorname{Pin}^+(d) \rtimes G_{int}}{\mathbb{Z}_2^F} \to \mathbb{Z}_2 \right).
$$
Here $(G_{int},(-1)^F,\theta_G)$ is an arbitrary fermionic symmetry group which acts on $\operatorname{Pin}^+(d)$ via the map $\theta_G: G_{int} \to \mathbb{Z}_2$ and the canonical automorphism of $\operatorname{Pin}^+(d)$. The combined map $\theta$ to $\mathbb{Z}_2$ is defined by $(x,g) \mapsto \theta_{\operatorname{Pin}}(x) \theta_G(g)$.
For example, for $G_{int} = \operatorname{Pin}^-(1)$, the internal symmetry group consisting of a single time-reversing symmetry $T$ such that $T^2 = (-1)^F$, the above group can be shown to be isomorphic to $\operatorname{Pin}^+(d)$.
For more details on Q2 and other examples, you could consider looking to section 2.1 and 3.2 of my preprint.
