First, in general, the eta invariant for a self-adjoint elliptic operator on closed manifold is not a topological invariant. It depends on the geometric structure of the manifold. For example in spin case, it depends on the spin structure. So we cannot expect that it can be writed only by characteristic numbers.
Next, the eta invariant is not local. In the second page of the original paper "Spectral Asymmetry and Riemannian Geometry I" by Atiyah-Patodi-Singer, they gave an nonlocal example about a suitable len space. So we cannot expect that it can be writed by an integral of some local terms.
So the result of Stolz is a surprising but not a general result.
As I know, if you want to write the eta invariant as a local expression or the spin structure, there are two point of views.
The first one followed by Atiyah-Patodi-Singer that if the manifold $M$ is the boundary of some manifold $W$, then the reduced eta invariant can be written as
$$\eta(M)=\mathrm{APS-index}(D^W)-\int_W \text{some characteristic class of}\ W.$$
So after mod $\mathbb{Z}$, we can regard the eta invariant of $M$ as an integral of local terms of $W$.
The second point of view is for the spin case. In this case, the eta invariant has a geometric expression that
$$\eta(M)=\frac{1}{\sqrt{\pi}}\int_0^{\infty}t^{1/2}\mathrm{Tr}[D\exp(-tD^2)]dt.$$
Here $D$ is the Dirac operator. So we can study the eta invariant using the heat kernel.
The Euler operator $d+d^*$ of a spin manifold is the spin Dirac operator
twisted by the $\mathbb Z/2$-graded spinor bundle $W=S^+\ominus S^-$, as explained in Heat Kernels and Dirac Operators by Berline, Getzler, Vergne [BGV, chapter 4.1]. The Chern character $\mathrm{ch}(W)$ equals $e(TM)\hat A(TM)^{-1}$, see [BGV] or Nicolaescu's notes. The cohomological index theorem therefore gives
$$\mathrm{ind}(d+d^*)^+=(\hat A(TM)\wedge\mathrm{ch}(W))[M]=e(TM)[M]$$
The de Rham representative of $e(TM)$ is given by the Pfaffian, and you get your second formula [BGV, Theorem 4.6].
The formulas above are pairings of "characteristic cohomology classes" with "fundamental homology classes". If you regard the Dirac operator as a fundamental $K$-homology class for the $K$-orientation of the tangent bundle given by the chosen spin structure, then the left hand side also constitutes a pairing of the $K$-theory class of the spinor bundle with the corresponding fundamental class.
To avoid the spin condition, you define the ``twist Chern character'' $\mathrm{tr}(\mathrm{exp}(-F^{W/S}))$ as explained in [BGV, chapter 4.1].
The formulas above still hold.
You can even get rid of orientability by regarding $e(TM)\in H^{\dim M}(M,o(TM))$ as a cohomology class twisted by the orientation bundle. Similarly, the Pfaffian is in fact a differential form twisted by the orientation bundle. Then all formulas above make sense without an orientation of $TM$.
On the other hand, the Euler number is the "trace" of id$_M$ in a sense explained nicely here (Def 2.2) - no matter if you count cells or dimensions of (co-) homology groups.
To relate it to the pairings above, you may use either Hodge theory, or the Gauss-Bonnet-Chern theorem, or deduce $e(TM)[M]=\chi(M)$ from the Poincar'e-Hopf theorem, e.g. using Morse theory. It seems that because $\chi(M)$ and $e(TM)[M]$ are different kinds of objects, some work is needed for this step.
Best Answer
1) The eta invariant itself depends on the metric, but the relative eta invariant is in many cases (see comments) a homotopy invariant. The relative eta invariant is defined to be the difference of the eta invariants associated to the Dirac operator twisted by two different flat Hermitian bundles (i.e. unitary representations of the fundamental group).
2) The relation between the eta invariant and Chern-Simons invariants is a little bit subtle, but it is explained in detail in section 4 of "Spectral Asymmetry and Riemannian Geometry II" by A-P-S.
3) Arguably the most important examples are lens spaces - this is how it was first realized that the defect in the signature theorem for manifolds with boundary is non-local, for example (if it were local it would be multiplicative for coverings). There is also an interesting paper called "Eta Invariants, Signature Defects of Cusps, and Values of L-Functions" by Atiyah, Donnelly, and Singer in which the eta invariant associated to the signature operator on a Hilbert modular variety with the cusps chopped off is calculated in terms of values of Shimazu L-functions. This was apparently one of the motivating examples for the theory of eta invariants, but I don't know what actual arithmetic significance it has.