Edit. After thinking about this problem a bit longer, I am not so sure anymore that the Bredon cohomology proposed by Adem and Ruan gives me the invariants I am looking for. I have therefore moved that part to the very bottom and changed some of the questions a bit.
Is there a definition of equivariant or orbifold Chern classes that restrict to $H^\bullet(-;\mathbb Z)$-valued Chern classes if the group $G$ is trivial, or if the orbifold is a smooth manifold in the sense that all isotropy groups are trivial?
Q1 How does one define a suitable orbifold cohomology theory $H^\bullet_{\mathrm{orb}}$ that becomes integral cohomology if all isotropy groups are trivial? The answer probably depends on compatible choices of $H^\bullet_{\mathrm{orb}}(\mathbb R^n/\Gamma)$ for all linear actions of finite groups.
Q2 Are there (natural etc.) Chern classes $c_k(V)\in H^\bullet_{\mathrm{orb}}(B)$?
For simplicity, assume we are dealing with an effective orbifold $B$. Then there exists a Lie group $G$ and a smooth manifold such that the orbifold is isomorphic to $M/G$. Note that $M$ and $G$ are not unique. A canonical choice for $M$ would be the orthogonal frame orbibundle of $TB$ (if $TB$ is given a Riemannian metric) and $G=O(n)$.
If $B=M/G$ then an orbifold vector bundle $V\to B$ can be identified by pullback along $p\colon M\to B$ with a $G$-equivariant vector bundle $p^*V$ on $M$.
Q3 Are there equivariant Chern classes $c_k(p^*V)\in H^\bullet_G(M)$ (where one should probably consider one of the definitions proposed in this question)?
Q4 Is there a (natural etc.) pullback map $p^*\colon H^\bullet_{\mathrm{orb}}(B)\to H^\bullet_G(M)$ such that $p^*c_k(V)=c_k(p^*V)$?
Note. I am particularly interested in torsion phenomena, so a $\mathbb Q$-version would not really help. On the other hand, I probably only need the non-equivariant part of $c_k(p^*V)$ (but vector bundles on $|B|$ instead of orbibundles will not suffice for my purpose).
To give some background, I've been reading Adem-Ruan's article on twisted orbifold $K$-theory. In Section 3, a cohomology theory is introduced, whose integral version would be defined as follows. Assume that a compact Lie group $G$ acts with finite stabilisers on a $C^\infty$-manifold $M$, so $B=G\backslash M$ is an orbifold. Assume that a $G$-CW structure on $M$ has been fixed. To each orbit type $G/H$, associate a cellular chain group $$C_\bullet^G(M)(G/H)=C_\bullet(M^H/WH_0)\;,$$ where $WH_0$ is the maximal connected subgroup of $N_G(H)/H$. Now, let $R(-)$ denote the complex representation ring and consider $$C_G^\bullet(M;R)=\mathrm{Hom}_{Or(G)}(C_\bullet^G(M),R)\subset\prod_H\mathrm{Hom}(C_\bullet(M^H/WH_0),R(H))\;,$$ where the subscript $Or(G)$ means that for each inclusion $K\subset H$, the homomorphisms should intertwine the pullback induced by $M^H/WH_0\to M^K/WK_0$ with the restriction $R(H)\to R(K)$. Then consider $$H_{\mathrm{orb}}(B)=H_G^\bullet(M;R)=H^\bullet(C_G^\bullet(M;R))\;.$$ Note that Adem and Ruan only consider everything $\otimes\mathbb Q$, which they show is independent of the choice of $M$ and $G$.