It is not true that $\int_{[M]^{vir}} 1 = 2875$ for all the moduli spaces you list. If we let $$N_n = \int_{[M(1,0,-1,n)]^{vir}} 1 $$
then
$$Z_1(q) = \sum_{n=1}^\infty N_n q^n = 2875\cdot M(-q)^{e(Y)}\cdot \frac{q}{(1+q)^2}$$
where
$$M(q) = \prod_{m=1}^\infty (1-q^m)^{-m}$$
What is going on here is that the moduli space $M(1,0,-1,n)$ is isomorphic to the Hilbert scheme of 1-dimensional subschemes $C$ in the class of a line with $\chi(\mathcal{O}_C)=n$. For $n=1$ this moduli space is 2875 points representing the 2875 lines, but for $n>1$, the subscheme will consist of a line and $n-1$ points (they could be far away from the lines or embedded into the lines). These moduli spaces are in general very complicated.
Note that by MNOP conjecture (recently proven for the quintic by Pandharipande-Pixton), the above formula for $Z_1(q)$ tells you what the degree 1 Gromov-Witten invariants are in all genus, not just genus 0.
So Donaldson-Thomas theory (the MNOP version using ideal sheaves of subschemes) is not a very efficient way of getting solely genus 0 information --- you have to compute for ideal sheaves of all euler characteristics, convert the corresponding (reduced) series in $q$ to a series in $\lambda$ via the substitution $q=-e^{i\lambda}$, pass from the disconnected to connected series, and then find the $\lambda^{-2}$ term in the expansion!
If you want purely genus 0 enumerative information from a Donaldson-Thomas like invariant, you could use Sheldon-Katz's proposed definition of genus 0 Gopakumar-Vafa invariants. He proposes
$$n_{0,\beta}(X) = \int_{[M(0,0,\beta,1)]^{vir}} 1$$
where $X$ is a Calabi-Yau threefold, $\beta \in H_2(X)$ is a curve class, and $M(0,0,\beta,1)$ is the moduli space of stable pure sheaves with chern character $(0,0,\beta,1)$. In your case above, this space will be populated by only the structure sheaves of the 2875 lines and so indeed the genus zero Gopakumar-Vafa of the quintic in the class of the line is 2875.
Of course, none of this really addresses your real question which is how do we get the number 2875 from Donaldson-Thomas theory without a priori knowing the enumerative answer. The answer is perhaps disappointing in this case. There is no good computational way to see the 2875 lines in DT theory. In Gromov-Witten theory, one can relate genus 0 invariants of $Y$ to Gromov-Witten like invariants of the ambient space $\mathbb{CP}^4$ and in the case of the 2875, this can be reduced to classical intersection theory. DT theory is only defined for threefolds and so there is no known way to related the DT theory of $Y$ to something on the ambient space. The only available computational technique here for the quintic is degeneration (degenerating the quintic to two fano threefolds meeting along a K3 for example). This is what is employed by Pixton-Pandharipande to prove the MNOP conjecture, but it is rather complicated and probably impossible to excise the genus zero part of the computation (for the reasons explained above). Conceivably, one could try to apply degeneration techniques to the Katz definition of genus 0 GV invariants, but that hasn't been explored and also probably isn't easy (one would be forced into considering moduli spaces with strictly semi-stable objects and that causes difficulty in DT theory).
One computational advantage that DT theory does have over GW theory is that it can be computed motivically: the integral over the virtual class is given by a weighted Euler characteristic and the euler characteristic of a space can be computed by summing the Euler characteristic of the strata in a stratification. This actually does provide a close connection between the DT theory (really PT theory --- DT theory's little brother) and the enumerative geometry of surfaces. The paper of Kool-Shende-Thomas use this idea to prove Gottsche's conjecture which gives a universal formula for the enumerative geometry of surfaces. The subsequent papers of Kool-Thomas make this connection between enumerative geometry of a surface $S$ and the PT theory of the threefold $K_S$ more explicit. Gottsche and Shende go further and use refined PT theory to obtain a refined version of Gottsche's conjecture which contains even subtler enumerative informations (like the number of real curves on a surface).
To summarize this long winded answer: DT theory is not very good at directly seeing the enumerative geometry of threefolds like the quintic (in particular the 2875 lines), but it is very good for understanding the enumerative geometry of surfaces.
Edit: here are some links to papers discussed above:
Pandharipande-Pixton prove the MNOP conjecture in a series of papers. This is the final one (which references the earlier ones)): http://arxiv.org/abs/1206.5490
Kool-Shende-Thomas proof of the Gottsche conjecture and subsequent papers by Kool-Thomas:
http://arxiv.org/abs/1010.3211
http://arxiv.org/abs/1112.3069
http://arxiv.org/abs/1112.3070
Gottsche-Shende's papers on refined Gottsche conjecture:
http://arxiv.org/abs/1208.1973
http://arxiv.org/abs/1307.4316
Brenin asked for a reference for the computation of $Z_1(q)$. Here is a direct computation in DT theory: http://arxiv.org/abs/math/0601203 , but it is easier to understand this computation in PT theory: see the example on page 28 in this beautiful expository paper:
http://arxiv.org/pdf/1111.1552.pdf
Best Answer
Let me give an answer from a slightly different point of view.
Let $M_k$ be a moduli space as in your question; say it's a (compact) moduli space of sheaves on some (compact) Calabi-Yau threefold. In general, $M_k$ is going to be very singular. However, it carries a so-called perfect deformation-obstruction theory of dimension zero. This gives a virtual fundamental class on $M_k$, and the technical definition of the numerical invariant $e(M_k)$ is that it is the degree of this virtual fundamental class.
In the case of sheaves on CY3s, the deformation-obstruction theory has a duality property: it is a symmetric obstruction theory. In this case, according to a result of Kai Behrend, $e(M_k)$ can also be expressed as an Euler characteristic, albeit a weighted one:
$e(M_k)=\chi(M_k, \nu_{M_k})$, where $\nu_{M_k}$ is the Behrend function of the singular space $M_k$. In other words, one computes an Euler characteristic, but weighted with a numerical measure of how bad the singularities are.
On can hope that this Euler characteristic definition can now be turned into something motivic. What one needs is a way to attach a motivic weight to points of $M_k$. In some specific moduli problems, such as for Hilbert schemes of points where at least locally the moduli space can be expressed as a critical locus of a function on a smooth variety, this can be done using the tool of the motivic vanishing cycle; indeed, this is what our work does in the paper you cite. The general theory of how one attaches motivic weights is discussed in a (partially conjectural) paper of Kontsevich and Soibelman.
The issue with Gromov-Witten theory on a CY3 is that the deformation-obstruction theory in that case, while it is of dimension zero, is not fully symmetric. It is symmetric on the open part corresponding to stable maps which are immersions from a smooth curve, but (as an expert assures me) not on the whole moduli space.