Q:

Computation of homology groups of $M_{g,n}$

First some definitions: $\bar{M_{g,n}}$ is Deligne-Mumford space, i.e., the moduli space of stable nodal complex projective curves of genus $g$ with $n$ marked points.  It is a complex orbifold, $\partial \bar{M_{g,n}}$ is the locus in $\bar{M_{g,n}}$ corresponding to nodal curves (with singularity).  Do we have any relationship between the homology groups $H_{*}(M_{g,n},Q)$  and the homology groups of $\bar{M_{g,n}}$, $\partial \bar{M_{g,n}}$, the pair $(\bar{M_{g,n}},\partial \bar{M_{g,n}})$ (relative homology); here $M_{g,n}$ is $\bar{M_{g,n}}\setminus \partial \bar{M_{g,n}}$ the locus of smooth curves? 
The point is for any pair of compact oriented manifolds $(X, Y), Y\subset X$, can we calculate the homology groups of $X\setminus Y$ from those of $X, Y$ and the relative homology groups $(X,Y)$ (it is not an excision case)? 
This is a problem I find on page 23 of the paper: Costello, "Gromov-Witten potential associated to a TCFT", (although there it is $\bar{M_{g,n}}/S_n$, modulo the action of permutation of marked points, but it is not a big deal). 
One more question is: is $\bar{M_{g,n}}/S_n$ orbifold? 

A:

There is some general machinery that is perfectly suited to this question. 
Consider the following setup:  let $U$ be the complement of a normal crossing divisor $D$ in a compact complex manifold (or orbifold) $X$.  (In the special case at hand, $U = \mathcal{M}_{g,n}$, and $X$ is the Deligne-Mumford compactification.)    With a bit of work one can see that the Leray spectral sequence for the inclusion $U\hookrightarrow X$ has $E_2$ page given by
\[
E^{p,q}_2 = \oplus_S H^{q-2p}(D_S;\mathbb{C})
\]
where the sum runs over the closed boundary strata of codimension $p$ and the differential on this page is given by $da = \Sigma_T \pm (i_{S,T})_!  a$  where $a\in H^{q-2p}(D_S;\mathbb{C})$ and $T$ runs over codimension $p-1$ boundary strata that contain $D_S$, and $(i_{S,T})_!$ is the pushforward along the inclusion $D_S \hookrightarrow D_T$.  One can also get this spectral sequence from the weight filtration on the complex of forms with logarithmic poles along $D$.
Deligne proved that this spectral sequence degenerates at the $E_2$ page.  So the cohomology of this page is the associated graded for $H^*(U;\mathbb{C})$.
In the case of the moduli space of curves and its Deligne-Mumford-Knudsen compactification, the $E_2$ page is described in terms of the cohomology of the various strata of the boundary, which are isomorphic to smaller compactified moduli spaces.   There have been quite a few papers that used this spectral sequence to prove interesting things.   For instance, there is an old paper of Voronov (alg-geom/9708019) (from just a couple of years before the proof of the Madsen-Weiss theorem) in which he analyzes this SS to show that the rational homotopy type of $\mathcal{M}_{g,n}$ is stable in the Harer-Ivanov stable range and moreover, it is formal in this range.   
As an aside, very shortly after Voronov's paper, there was Tillmann's paper in which she showed that $\mathcal{M}_{g,n}$ has the homology of an infinite loop space in the stable range, which also implies rational homotopy formality in the stable range.