Q1: Let me first note, that the statement $$H^\ast(BG,\mathbb{Q}) \cong H^\ast(BT,\mathbb{Q})^W\tag{$\ast$}$$ made in the question requires in addition that $G$ is connected. However, the general case can be reduced to the connected case since $$H^\ast(BG,\mathbb{Q}) \cong H^\ast(BG_0,\mathbb{Q})^{G/G_0}$$ where $G_0$ is the identity component of $G$.

A text book reference for $(\ast)$ can be found in [Hsiang: Cohomology theory of topological transformation groups, Chapter III, §1, Lemma 1.1]. The results of the book that are relevant for your question can also be found in the following paper: http://www.math.uwo.ca/~rgonzal3/qfy.pdf (cf. Remark 9, Lemma 5). Other approaches and more information can be found in

$\quad$cohomology of BG, G compact Lie group

Q2: First note that the kernel of the restriction map $$\rho^\ast: H^\ast(BG;\mathbb{Z}) \to H^\ast(BT;\mathbb{Z})$$ is the tosion subgroup $Tors$ of $H^\ast(BG;\mathbb{Z})$. So, $\rho^\ast$ is injective iff $H^\ast(BG;\mathbb{Z})$ is torsion-free (there is a lot of literature about torsion of $H^\ast(BG;\mathbb{Z})$ so I won't discuss it here).

A short paper of Feshbach [The image of $H^\ast(BG;\mathbb{Z})$ in $H^\ast(BT;\mathbb{Z})$ for $G$ a compact Lie group with maximal torus $T$. Topology 20(1981),93-95] characterizes when the induced map $$\bar{\rho}^\ast: H^\ast(BG;\mathbb{Z})/Tors \to H^\ast(BT;\mathbb{Z})$$ is an isomorphism:

$\quad\bar{\rho}^\ast$ is an isomorphism iff $H^\ast(BG;\mathbb{Z})/Tors \otimes \mathbb{F}_p$ is an integral domain for each prime $p$.

There is also a counter-example that shows that $\bar{\rho}^\ast$ is not always an isomorphism.

Q1: Let me first note, that the statement $$H^\ast(BG,\mathbb{Q}) \cong H^\ast(BT,\mathbb{Q})^W\tag{$\ast$}$$ made in the question requires in addition that $G$ is connected. However, the general case can be reduced to the connected case since $$H^\ast(BG,\mathbb{Q}) \cong H^\ast(BG_0,\mathbb{Q})^{G/G_0}$$ where $G_0$ is the identity component of $G$.

A text book reference for $(\ast)$ can be found in Hsiang, Cohomology theory of topological transformation groups, Chapter III, §1, Lemma 1.1. The results of the book that are relevant for your question can also be found in the following paper: Richard Gonzales, Localization in equivariant cohomology and GKM theory (cf. Remark 9, Lemma 5). Other approaches and more information can be found in  this answer.

Q2: First note that the kernel of the restriction map $$\rho^\ast: H^\ast(BG;\mathbb{Z}) \to H^\ast(BT;\mathbb{Z})$$ is the tosion subgroup $Tors$ of $H^\ast(BG;\mathbb{Z})$. So, $\rho^\ast$ is injective iff $H^\ast(BG;\mathbb{Z})$ is torsion-free (there is a lot of literature about torsion of $H^\ast(BG;\mathbb{Z})$ so I won't discuss it here).

A short paper of Feshbach

• The image of $H^\ast(BG;\mathbb{Z})$ in $H^\ast(BT;\mathbb{Z})$ for $G$ a compact Lie group with maximal torus $T$. Topology 20(1981) 93-95 doi:10.1016/0040-9383(81)90015-X,

characterizes when the induced map $$\bar{\rho}^\ast: H^\ast(BG;\mathbb{Z})/Tors \to H^\ast(BT;\mathbb{Z})$$ is an isomorphism:

$\bar{\rho}^\ast$ is an isomorphism iff $H^\ast(BG;\mathbb{Z})/Tors \otimes \mathbb{F}_p$ is an integral domain for each prime $p$.

There is also a counter-example, $\mathrm{Spin}(12)$, that shows that $\bar{\rho}^\ast$ is not always an isomorphism.

Q1: Let me first note, that the statement $$H^\ast(BG,\mathbb{Q}) \cong H^\ast(BT,\mathbb{Q})^W\tag{$\ast$}$$ made in the question requires in addition that $G$ is connected. However, the general case can be reduced to the connected case since $$H^\ast(BG,\mathbb{Q}) \cong H^\ast(BG_0,\mathbb{Q})^{G/G_0}$$ where $G_0$ is the identity component of $G$.

A text book reference for $(\ast)$ can be found in [Hsiang: Cohomology theory of topological transformation groups, Chapter III, §1, Lemma 1.1]. The results of the book that are relevant for your question can also be found in the following paper: http://www.math.uwo.ca/~rgonzal3/qfy.pdf (cf. Remark 9, Lemma 5). Other approaches and more information can be found in

$\quad$cohomology of BG, G compact Lie group

Q2: First note that the kernel of the restriction map $$\rho^\ast: H^\ast(BG;\mathbb{Z}) \to H^\ast(BT;\mathbb{Z})$$ is the tosion subgroup $Tors$ of $H^\ast(BG;\mathbb{Z})$. So, $\rho^\ast$ is injective iff $H^\ast(BG;\mathbb{Z})$ is torsion-free (there is a lot of literature about torsion of $H^\ast(BG;\mathbb{Z})$ so I won't discuss it here).

A short paper of Feshbach [The image of $H^\ast(BG;\mathbb{Z})$ in $H^\ast(BT;\mathbb{Z})$ for $G$ a compact Lie group with maximal torus $T$. Topology 20(1981),93-95] characterizes when the induced map $$\bar{\rho}^\ast: H^\ast(BG;\mathbb{Z})/Tors \to H^\ast(BT;\mathbb{Z})$$ is an isomorphism:

$\quad\bar{\rho}^\ast$ is an isomorphism iff $H^\ast(BG;\mathbb{Z})/Tors \otimes \mathbb{F}_p$ is an integral domain for each prime $p$.

There is also a counter-example that shows that $\bar{\rho}^\ast$ is not always an isomorphism.

Q1: Let me first note, that the statement $$H^\ast(BG,\mathbb{Q}) \cong H^\ast(BT,\mathbb{Q})^W\tag{$\ast$}$$ made in the question requires in addition that $G$ is connected. However, the general case can be reduced to the connected case since $$H^\ast(BG,\mathbb{Q}) \cong H^\ast(BG_0,\mathbb{Q})^{G/G_0}$$ where $G_0$ is the identity component of $G$.

A text book reference for $(\ast)$ can be found in [Hsiang: Cohomology theory of topological transformation groups, Chapter III, §1, Lemma 1.1]. The results of the book that are relevant for your question can also be found in the following paper: http://www.math.uwo.ca/~rgonzal3/qfy.pdf (cf. Remark 9, Lemma 5). Other approaches and more information can be found in

$\quad$cohomology of BG, G compact Lie group

Q2: First note that the kernel of the restriction map $$\rho^\ast: H^\ast(BG;\mathbb{Z}) \to H^\ast(BT;\mathbb{Z})$$ is the tosion subgroup $Tors$ of $H^\ast(BG;\mathbb{Z})$. So, $\rho^\ast$ is injective iff $H^\ast(BG;\mathbb{Z})$ is torsion-free (there is a lot of literature about torsion of $H^\ast(BG;\mathbb{Z})$ so I won't discuss it here).

A short paper of Feshbach [The image of $H^\ast(BG;\mathbb{Z})$ in $H^\ast(BT;\mathbb{Z})$ for $G$ a compact Lie group with maximal torus $T$. Topology 20(1981),93-95] characterizes when the induced map $$\bar{\rho}^\ast: H^\ast(BG;\mathbb{Z})/Tors \to H^\ast(BT;\mathbb{Z})$$ is an isomorphism:

$\quad\bar{\rho}^\ast$ is an isomorphism iff $H^\ast(BG;\mathbb{Z})/Tors \otimes \mathbb{F}_p$ is an integral domain for each prime $p$.

There is also a counter-example that shows that $\bar{\rho}^\ast$ is not always an isomorphism.