I once heard Witten say that topology in 5 and higher dimensions "linearizes". What he meant by that is that the geometric topology of manifolds reduces to algebraic topology. Beginning with the Whitney trick to cancel intersections of submanifolds in dimension $d \ge 5$, you then get the h-cobordism theorem, the solution to the Poincare conjecture, and surgery theory. As a result, any manifold in high dimensions that is algebraically close enough to $\mathbb{R}^d$ is homeomorphic or diffeomorphic to $\mathbb{R}^d$.
By the work of Freedman and others using Casson handles, there is a version of or alternative to the Whitney trick in $d=4$ dimensions, but only in the continuous category and not in the smooth category. Otherwise geometric topology does not "linearize" in Witten's sense. But in $d \le 3$ dimensions, the dimension is too low for the smooth category to separate from the continuous category, at least for the question of classification of manifolds.
What you have in 3 dimensions is examples such as the Whitehead manifold, which is contractible but not homeomorphic to $\mathbb{R}^3$. In 4 dimensions you instead get open manifolds that are homeomorphic to $\mathbb{R}^4$ (because they are contractible I'm not sure if other conditions are needed and simply connected at infinity), but not diffeomorphic to $\mathbb{R}^4$. You have to be on the threshold between low dimensions and high dimensions to have the phenomenon. I would say that these exotic $\mathbb{R}^4$s don't really look that much like standard $\mathbb{R}^4$, they just happen to be homeomorphic. The homeomorphism has fractal features, and so does the Whitehead manifold.
Meanwhile 2 dimensions is too low to have non-standard contractible manifolds. In the smooth category, the Riemann uniformization theorem proves that smooth 2-manifolds are very predictable, or you can get the same result in the PL category with a direct combinatorial attack on planar graphs. And as mentioned, smooth, PL, and topological manifolds don't separate in this dimension.
Also, concerning your question about Cartesian products: Obviously the famous results imply that there is a fibration of standard $\mathbb{R}^5$ by exotic $\mathbb{R}^4$. The Whitehead manifold cross $\mathbb{R}$ is also homeomorphic to $\mathbb{R}^4$. (I don't know if it's diffeomorphic.) These fibrations are also fractal or have fractal features.
Given a paracompact smooth manifold, you have smooth partitions of unity (nLab), but on a real analytic manifold (e.g. a complex manifold viewed as a real manifold) one doesn't have analytic partitions of unity (much less holomorphic, if you are in the complex case). That is, given any open cover on a smooth manifold, one can find a partition of unity subordinate to that cover - this is a very topological property. Using partitions of unity you can paste together local functions as desired.
The existence of smooth partitions of unity comes down to the existence of a smooth (but not analytic!) bump function on $[-1,1]$. Edit: you can find details and formulas on Wikipedia.
A related fact is that on a paracompact smooth manifold, the sheaf of real-valued functions is fine (nLab,wikipedia).
Best Answer
There are two possible meanings for the sentence "f : M → N admits local sections", so let's first disambiguate.
Meaning 1: For every point of N, there exists a neighborhood of that points and a section from that neighborhood back to M.
That's what people typically check in order to verify that, say, a map is a $G$-principal bundle.
Meaning 2: For every point m ∈ M, there exists a neighborhood of $f(m)$, and a section s from that neighborhood back to M, subject to the extra condition that $s(f(m))=m$.
Clearly, you care about the second meaning of that sentence.
It is correct that a map is a submersion (not necessarily surjective!) iff it admits local sections.
If a map has local sections, then the maps on tangent spaces are sujective: that's just obvious.
Conversely, if a map is surjective at the level of tangent spaces, you first pick a local section of the maps of tangent spaces. Then, to finish the argument, you use the fact that any subspace of the tangent space $T_mM$ is the tangent space of a submanifold of M, and apply the implicit function theorem.
Note: if you care about infinite dimensional Banach manifolds, then the existence of a section for the map to tangent spaces needs to be assumed a separate condition. Indeed, it's not enough to assume that the map of tangent spaces is surjective, since it's not true that any surjective map of Banach spaces has a section.
Note: For complex varieties, you don't have the implicit function theorem, so it doens't work. Counterexample: the map $z\mapsto z^2$ from ℂ* to itself. The fix is to pass the the "étale topology"... but that's another story.