Confusion about induced bundles, equivariant Vector bundles and representations

Question

I am writing this question to try to unconfuse myself in two different directions.

Throughout we work over $\mathbb{C}$ and the choice of topology should not matter, but say we are using the etale topology. Let $G$ be an algebraic group, $X$ an algebraic variety over which $G$ acts. I want to study equivariant Vector bundles on $[X/G]$.

1. My First question is this: I know that the Vector bundles on X are locally free sheaves. The Vector bundles on the quotient are defined to be the G-equivariant among these ones, right?

Then I want to understand the case of a single point. The equivariant, locally free sheaves of finite rank should correspond to complex representation of the group of $\mathbb{C}$-points of $G$. Now, if $G$ was a complex Lie group and I was thinking of actual vector bundles this would be clear. Here instead, following the definition of equivariant sheaf in the stacks project 043T, I need an isomorphism alpha between the pullback of the sheaf to $G\times_{\text{Spec}(\mathbb{C})} \text{Spec}(\mathbb{C})\cong G$ through the action map and the pullback of the sheaf through the projection map.

Now, a locally free sheaf with finite rank $k$ over a point is just a $k$-dimensional vector space and pulling it back, no matter the map, I get $k$ copies of the structure sheaf of $G$.

2. Are such isomorphisms in bijection with the complex representations of $G$? If so, how? What is the dictionary/ could anyone give me a reference for it?

3. Now, let me pullback any such equivariant sheaf from the point to my variety X. What I get is always an equivariant sheaf, but this should be globally free, right?

4. In differential geometry there is an equivalence between smooth finite dimensional vector bundles and localy free sheaves of finite rank. Is there also an equivalence between equivariant finite dimensional vector bundles and equivariant, locally free sheaves of finite rank? If yes, how do I get the isomorphism as in point 2, which is the datum required in the definition of equivariant sheaf?

I hope that the points of the question make clear the ways and the effort I am putting to approach the "problem", even though this is mostly uncertainty about new notions.