This is a layman's answer.
What you are probably after is something like mean curvature flow. This evolution equation does concentrate curvature -- a good reference for this happening with convex initial hypersurfaces is Huisken's seminal 1984 paper in JDG. More recently, people have worked on surgery for the mean curvature flow, which does break the hypersurface into distinct regions where the curvature concentrates into singularities.
For your specific question, it isn't possible. Uniqueness in backward time is rubbish: the PDE is not parabolic. There is a clear and intuitive reason for this. The Ricci flow evolves geometries toward a model state: on a sphere with positive curvature, toward the geometry of a sphere. Now think about this for a second. All positive curvature geometries evolve toward one geometry -- as you wrote, the effect of the Ricci flow is to `smooth out' regions of curvature. Now if I begin at the goal model geometry, and flow in the backward time direction, how can I possibly know which less-ideal-but-positively-curved geometry to pick?
This is the basic problem with equations which aren't parabolic. In general, to reverse time is a very special and tricky thing and you need a lot more information to do it with any meaning. In this sense, I'd venture that the Ricci flow isn't the tool you're looking for.
$\newcommand{\R}{\mathbb{R}}$
The basic heat equation on a domain in $\R^n$ is
$$ \partial_tu = \sum_{i=1}^n (\partial_i)^2u $$
The inhomogeneous version is
$$ \partial_tu = \sum_{i=1}^n (\partial_i)^2u + f, $$
where $f$ is a function of $x \in \R^n$ only.
This all can be generalized to the linear PDE
$$ \partial_tu = a^{ij}(x,t) \partial^2_{ij}u + b^k(x,t)\partial_ku + c u+ f(x), $$
where the matrix $A = [ a_{ij}]$ is positive definite.
This is the most general version of what is often called a linear heat equation on a domain in $\R^n$.
A quasilinear heat equation is simply one where the functions $a^{ij}$, $b^k$, $c$, and $f$ are functions of not only $x$, $t$, but also $u$.
On a Riemannian manifold, one possible linear heat equation is of the form
$$
\partial_tu = \Delta_g u + b^k\partial_k u + cu + f,
$$
where $\Delta_g u = g^{ij}\nabla^2_{ij}u$.
All of the above assumes $u$ is a scalar function on its domain.
Where things get more complicated is when $u$ is either a map $u: M \rightarrow N$, as it is for the harmonic map heat flow or the Riemannian metric itself, as it is for the Ricci flow. If you write out the formulas for these PDEs in local coordinates, you will see that the coefficients of the PDE depend on the unknown map or metric and therefore the PDE is nonlinear. The fact that the harmonic map heat flow can be written simply as
$$
\partial_t u = \Delta u
$$
is misleading, because here $\Delta u$ is a nonlinear function of its first and second partial derivatives.
As for why the Ricci flow is a nonlinear heat equation is longer story, but after you use the so-called DeTurck trick, the Ricci flow looks like
$$
\partial_t g_{ij} = g^{pq}\partial^2_{pq}g_{ij} + \text{ lower order terms}
$$
which is a nonlinear heat equation for the metric $g$.
There are similar stories for other geometric heat flows, such as the mean curvature flow.
Best Answer
When $n=3$, the seminal work of Hamilton shows that any closed 3-manifold with positive Ricci curvature converges to the constant curvature metric under the (normalized) Ricci flow. When $n=4$, Hamilton also has a similar result, assuming that the closed manifold has positive curvature tensor. More recently Bohm and Wilking show in this paper the same conclusion holds if the manifold has 2-positive curvature operator in any $n\ge 3$. Building on the idea of Bohm and Wilking, Brendle and Schoen use Ricci flow to prove the differentiable sphere theorem.
In general, as the other answer suggests, singularities develop under Ricci flow even when $n=3$. For example, Angenent and Knopf construct explicit examples of type I singularity for Ricci flow in the sphere in any dimension. More complicated singularities (the so called degenerate neckpinches) are later constructed by Angenent, Isenberg and Knopf.