This is rather late, and not a full answer to your question, but may be interesting nonetheless.
Concerning point 1), I would like to propose the following natural refinement which avoids the problems mentioned in Pavel Etingof's answer. If you want to know the relation between cohomology of the group and the Lie algebra over $\mathbb{Q}$, you should work with $\mathbb{Q}$-forms of both. Take $G_{\mathbb{Q}}$ a form of $G$ defined over $\mathbb{Q}$, and take $\mathfrak{g}_{\mathbb{Q}}$ the associated Lie algebra (in the sense of algebraic groups). Then you want to compare the algebraic de Rham cohomology of $G_{\mathbb{Q}}$ with the Lie algebra cohomology of $\mathfrak{g}_{\mathbb{Q}}$. As Pavel Etingof pointed out, these $\mathbb{Q}$-vector subspaces of the cohomology of $G$ over $\mathbb{C}$ depend on the choice of form. Maybe there is an explicit formula which from a cocycle in $H^1_{\text{ét}}(\mathbb{Q},\operatorname{Aut}G)$ produces the factors relating the subspaces - the factor $i$ in Pavel Etingof's answer is most certainly related to defining $SU(3)$ in terms of the complex conjugation involution.
Concerning point 2), the same as in point 1) applies to $\mathbb{Z}$-forms: choose a $\mathbb{Z}$-form of $G$ and take its associated Lie algebra. You will most likely end up getting Kostant's $\mathbb{Z}$-forms mentioned in Scott Carnahan's comment.
In the special case $GL_n$ and cohomology in top degree, your question 2) and 3) are discussed in a preprint of Annette Huber and Wolfgang Soergel, arXiv version can be found here. The goal of that paper is to compare different integral structures in the top cohomology of $GL_n$: one integral structure comes from the natural $\mathbb{Z}$-form of $\mathfrak{gl}_n$, another from the suspensions of Chern classes in de Rham cohomology, and the third one from the dual of the fundamental class in singular cohomology. They work out the explicit comparison factors. The comparison between de Rham and singular cohomology involves the usual period $2\pi i$, and the comparison between de Rham cohomology and the subspace coming from the Lie group cohomology is in fact a rational factor (so no further periods).
The rational factor in the comparison between Lie algebra and de Rham cohomology that is worked out in the Huber-Soergel paper is $\prod_{j=1}^n(j-1)!$. It is interesting to note that these same factors appear when comparing homotopy and homology of $GL_n(\mathbb{C})$ - the Hurewicz map sends a generator of $\pi_{2j-1}GL_n(\mathbb{C})$ to some multiple of a generator of $H_{2j-1}(GL_n(\mathbb{C}),\mathbb{Z})$, and this multiple is $(j-1)!$. I am fairly convinced now that this is the way the subspaces are related in case $GL_n$ - the group cohomology classes in degree $2j-1$ should be the $(j-1)!$-th multiple of the generators of Lie algebra cohomology. One would hope that such a description applies to other Lie groups and that the rational factor can be explained by some Weyl group combinatorics in general, but there is some work to be done for that...
Concerning periods there is a philosophical reason why the comparison between de Rham cohomology and singular cohomology should only involve rational multiples of powers of $2\pi i$. If $G$ is a reductive group over $\mathbb{Q}$, we can view it as a variety and as such it has a mixed Tate motive; therefore, all periods should be rational multiples of powers of $2\pi i$.