How do manifolds have enough structure to do calculus? I am referring, of course, to to differentiable manifolds. I've seen a few different definitions. The one I like best is the one which says it's a topological space such that every point has a neighborhood homeomorphic to an open set in $\mathbb{R}^n$ and the overlap functions are diffeomorphisms.
I can see that we want the transition maps are diffeomorpisms, so the calculus we do on each coordinate chart agrees with the others, but how do homeomorphisms to Euclidean space give enough structure to develop a derivative, for instance, in a coordinate patch? I mean, the circle and the sphere are homeomorphic but I'd only call one of those smooth. How does smoothness on overlaps do it for the whole manifold? I've seen it said that once a differentiable structure is defined, the homeomorphisms become diffeomorphisms. Is this correct and could somebody expand on that if it is? 
Furthermore, I've seen definitions given for manifolds with bijections rather than homeomorphisms. Can that possibly work? Bijections don't have to be particularly nice at all, so I can't for the life of me see how you'd get enough structure out of that. Please correct me if I've misunderstood some part of this.
 A: You don't do calculus directly on the manifold. Everything is done in the world of coordinates. 
Your coordinate maps let you push things down into $\mathbb{R}^n$ where we can take limits, compute partials, find Jacobian matrices etc.
Requiring the coordinate maps to be homeomorphisms is redundant. Why? Take your manifold to be a set equipped with some altas (bijective coordinate charts). Then you just declare that these coordinate charts (bijections) are homeomorphisms. In that way you force a topology on the set and turn in into a topological space (now equipped with homeomophism coordinate charts).
The trick to manifold theory is that you're never really working directly on the manifold itself. You always translate to the world of coordinates and work there.
Edit: To address @studiosus comment...
Fair enough, my last paragraph is a bit overstated. Maybe I should say instead...
To do concrete "calculus" computations, you must (usually) translate to the world of coordinates and work there. This is much like it is in geometry. You can do a lot with synthetic geometry, but when you want to calculate...add coordinates and start computing.
A: Given a smooth structure on the manifold $M$ the homeomorphisms become diffeomorphisms. To see that let $\varphi_\alpha:U_\alpha\subset M\to\mathbb{R}^n$ be charts on $M$. A map $f:M\to M$ is said to be smooth, if its local representation $\varphi_\alpha\circ f\circ\varphi_\beta^{-1}:\mathbb{R}^n\to \mathbb{R}^n$ is smooth. Now consider the local representation of a homeomorphism $\varphi_\alpha$ and note that smoothness follows by definition of the smooth structure (since the transition maps $\varphi_\alpha\circ\varphi_\beta^{-1}$ are smooth). Thus, $\varphi_\alpha$ indeed is a diffeomorphism on $M$. 
