Is there a topological description of combinatorial Euler characteristic?
the best approach to the geometric euler characteristic comes from the theory of o-minimal structures.
the best reference in this area is the book "tame topology and o-minimal structures" by lou van den dries. requires very little background to understand.
in brief: an o-minimal structure is collection of boolean algebras of subsets of $R^n$ which satisfies a short list of axioms. (the name comes from model theory, but you don't need to know any model theory to understand the results)
examples of o-minimal structures include the semialgebraic sets, the globally subanalytic sets, and (if you tweak the definitions a bit) the piecewise-linear sets.
elements of an o-minimal structure are "tame" or "definable" sets. mappings between tame sets are tame iff their graph is a tame set.
basic relevant results:
every tame set has a well-defined euler characteristic.
two tame sets are "definably homeomorphic" (there is a tame bijection between them --- not necessarily continuous!) iff they have the same dimension and euler characteristic.
(yes, i wrote iff - this is the first surprise in this subject)
one can so this for more general manifolds as well.
concerning integration with respect to euler characteristic:
1) in the o-minimal framework, one can integrate all constructible functions, as noted by viro and schapira in the 1980s, based on works of macpherson and kashiwara in the 1970s. these results follow from sheaf theory. though more difficult than the combinatorial approach, all these proofs are "natural" and don't rely on "luck".
2) if you want to integrate non-constructible (e.g., smooth) integrands, the theory of chen (really due to rota) will fail -- that integral vanishes on all continuous integrands.
3) baryshnikov and ghrist have extensions of the integral to definable integrands (see 2009 arxiv paper). there are two such extensions, and they are dual. there are deep connections with morse theory, but the integral operators are unfortunately non-linear, and the fubini theorem does not hold in full generality.
Why isn't there an intrinsic topological description, or perhaps manifold-theoretic description?
At least in some cases, the combinatorial Euler characteristic of X is equal to the homotopy Euler characteristic of the one-point compactification of X minus 1. For instance this is true when X is compact (of course) and also when X = R^n. It's true for all "nice" subsets of R^1. I don't know whether it works when X is, say, the open unit square plus one of its vertices.
Of course the first question is whether the combinatorial Euler characteristic is even a homeomorphism invariant. I would like to know the answer also.
a few comments on the comments above:
the combinatorial euler characteristic of a definable space (roughly speaking, a space with finite decomposition into finite-dimensional cells) is a homeomophism invariant, but not a homotopy invariant. there is a corresponding homological definition in terms of borel-moore homology or, if preferred, cohomology with local coefficients. it is, indeed, an invariant of definable bijections --- which do not need to be continuous.
the "polyconvex" sets are the combinatorial approach to "tame" sets. if you use the o-minimal theory, then you can vastly generalize the class of spaces for which euler characteristic is well-defined.
one reason to use the combinatorial (sometimes called "geometric") euler characteristic is that it satisfies the mayer-vietoris principle (or inclusion/exclusion) without requiring the spaces to be compact. specifically, $\chi(A \cup B) = \chi(A) + \chi(B) - \chi(A \cap B)$. hence, you can treat $d\chi$ as a finitely-additive signed measure on definable spaces and integrate constructible functions.
i apologize for harping on the o-minimal theory, but i found that it greatly simplified and generalized otherwise clunky proofs. the book of van den dries on the subject is very elementary and clear.