That's an old post but It got me really interested. What do you have in mind in (1)? What cohomology do you mean? Lie algebra cohomology?

Together with ACL's answers my concerns are completely covered. Thanks!

While it does answer my question, my actual problem is I don't understand what my construction actually gives if not affine schemes. Over affine schemes it obviously holds, what happens more generally?

Thanks! What about the contra-equivalence? Why doesn't it work?

This is exactly what I wanted thanks.The part with $Aut(P)$ was me incorrectly involving the global gauge group in the discussion. Thanks! Where would you recommend to read about all this homotopical algebraic perspective on fibre bundles?

@QiaochuYuan Thanks, That was enough for what i needed.

@QiaochuYuan Got it! Is there a direct way to see why this implies the existence of a flat connection?

@QiaochuYuan "Lift" means path lifting here?

whis was very helpful, thanks. One thing still bugs me though. Suppose I choose a connection for the bundle classified by $f$. This connection gives a holonomy representation $\Omega X \to Aut(P) \subset G$ into the gauge group of the bundle. What's the relation, if there's any at all, between this map and map $\Omega f : \Omega X \to G$? (I know the second map is unique only upto homotopy but stil...)

Thanks! What's the name of the "theory" that concerns itself with the kind of "topological monodromy" i'm talking about? Just so I could find my way in the literature

What does "flat" mean if your bundle has no smooth structure?