Stacks live in a $2$-category, so automorphisms of a stack form a $2$-group. For $BG$ the automorphism $2$-group should (maybe this needs some hypotheses on the field $k$) be a certain $2$-group built out of the conjugation action of $G$ on $\text{Aut}(G)$, satisfying $\pi_0 \cong \text{Out}(G)$ and $\pi_1 \cong Z(G)$.

Wikipedia does not require that last condition; it says "where possible..."

You can just apply the primitive element theorem to a finite extension of $\mathbb{R}(x)$. There are proofs that make this algorithmic, e.g. you can induct on the fact that if $\alpha, \beta$ are algebraic over $K$ then the set of $t \in K$ such that $\alpha + t \beta$ is not a primitive element of $K(\alpha, \beta)$ is the set of roots of a polynomial you can explicitly compute, and in particular there are finitely many such $t$.

The universal motivic measure is, tautologically, the identity map to $K_0$ itself. If you want a universal motivic measure which satisfies some further relations then you quotient $K_0$ by those relations. Of course this isn't all that useful.

Are you talking about the representation ring (en.wikipedia.org/wiki/Representation_ring)? This is completely understood for the reductive groups. For $GL_n$ it is isomorphic in a fairly explicit way to the ring of symmetric functions in $n$ variables, where the isomorphism is given by taking the character thought of as a function on diagonal matrices, and in particular it is finitely presented and has Krull dimension $n$. In general it is isomorphic (over $\mathbb{C}$) to $S(\mathfrak{h}^{\ast})^W$ where $\mathfrak{h}$ is a Cartan subalgebra and $W$ is the Weyl group.

@Simon: I was under the impression that this does not actually give a proof of Bott periodicity, and one has to use Bott periodicity itself to relate Clifford algebras to Bott periodicity?