Hey Theo --- I don't think it is reasonable to expect Lagrangian fibrations to be cotangent bundles globally. Easy example: take a 2d torus, give it a symplectic form (equivalently a volume form in this case); every 1d submanifold is automatically Lagrangian; the torus is a circle bundle over a circle; realizing it this way, it is a fibration over the circle with fibers being Lagrangian circles. Certainly this is not a cotangent bundle.
However, there is Weinstein's theorem, which says that given a Lagrangian and a point on it, there's a neighborhood of that point that looks like a cotangent bundle and such that in that neighborhood that Lagrangian looks like the zero section (or a fiber) of that cotangent bundle. Thus a Lagrangian fibration looks like a cotangent bundle in the way that you want locally. Sorry if this is confusing, if you want I can rewrite it in a less informal way. Anyway the answer to question 1 is yes, by Weinstein.
The above paragraph only explains the local picture; in symplectic geometry the local picture usually doesn't tell us much about the global picture.
Another example, integrable systems yield Lagrangian fibrations over R^n: these are usually not cotangent bundles. See the section on integrable systems in Cannas da Silva's book.
Edit: Sorry, I think I misread your question and took it to be more of a global question than it actually is.