About the rationality of contractible varieties: **Yes** for curves and surfaces and is an **open question** for higher dimensions. 

Any such contractible variety $X$ has $\chi_{top}(X)=1$, obviously. 

If $X$ is a curve then it must have only cusps as singularities, if any, by a simple $\chi_{top}$ calculation. Now let $Y$ be a projective model of $X$ such that it is smooth at the points in $Y-X$. Topologically, $Y$ is a real surface without boundary such that a few punctures make it contractible. The only real surface with this property is $S^2$, obviously. Hence $Y$ better be rational and so is $X$.   

If $X$ is an algebraic surface then it was a conjecture of Van de Ven that such a surface must be rational (actually his conjecture is for any homologically trivial $X$). This was proved by Gurjar & Shastri in: 

* *[On the rationality of complex homology 2-cells][1]*
* Here is the [the part II][2] of the above paper (MathSciNet review number MR0984747)

 [1]: https://dx.doi.org/10.2969/jmsj/04110037
 [2]: https://projecteuclid.org/journals/journal-of-the-mathematical-society-of-japan/volume-41/issue-2/On-the-rationality-of-complex-homology-2-cells-II/10.2969/jmsj/04120175.full