Topologically contractible algebraic varieties

No. Counterexamples were first constructed by Winkelmann, as quotients of $\mathbb A^5$ by algebraic actions of $\mathbb G_{\text{a}}$. I learned this from Hanspeter Kraft's very nice article available here:

http://www.numdam.org/numdam-bin/item?id=SB_1994-1995__37__295_0.

Recently Aravind Asok and Brent Doran have been studying these kinds of examples in the setting of $\mathbb A^1$-homotopy theory, on the arxiv as math/0703137.

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
• Here is the the part II of the above paper (MathSciNet review number MR0984747)