Contractible manifold with boundary - is it a disc?

If $M$ is contractible and the boundary of $M$ is simply-connected and $n\ge 6$ then $M$ is diffeomorphic to $D^n$. See Milnor's "Lectures on the h-cobordism theorem".

Sergei, there are lots of compact contractible smooth manifolds; see e.g. my answer here.

I am a bit confused about what you say next. Are you claiming that any compact contractible manifold admits the metric as you describe?

You might be interested in a paper of Ancel-Guilbaut who put a negatively curved (in the comparison sense) metric on the interior of any compact contractible manifold; see also discussion of this paper on the bottom of page 4 of the paper by Alexander-Bishop here.

Given a function $\psi:\mathbb R\to \mathbb R$, set $$\Psi=\psi\circ\mathrm{dist}_ {\partial M},\ \ \ \ \ f=\Psi\cdot(R-\mathrm{dist}_ p)$$ for some fixed $R>\mathrm{diam}\ M$.

Further, $$d\,f = (R-\mathrm{dist}_ p)\cdot d\,\Psi-\Psi\cdot d\,\mathrm{dist}_ p$$

Thus, we may choose smooth increasing $\psi$, such that $\psi(0)=0$ and it is constant outside of little nbhd of $0$ so that $\Psi$ is smooth. (It is possible since the function $\mathrm{dist}_ {\partial M}$ is smooth and has no critical points in a small neighborhood of $\partial M$.) Note that $d\,\Psi$ is positive muliple of $d\,\mathrm{dist}_ {\partial M}$. Thus $d_x\,f=0$ means that geodesic from $x$ to $p$ goes directly in the direction of minimizing geodesic from $x$ to $\partial M$, which can not happen.

Now we can apply Morse theory for $f$...