The answer is yes, to both questions.
First question first. For any geodesic $n$-gon $P$ on $M$, i.e., a simply connected region of $M$ whose boundary consists of $n$-geodesic arcs, define
$$ \delta(P)= \mbox{sum of the angles of $T$}-(n-2)\pi. $$
Note that if $M$ were flat, then the defect $\delta(P)$ would be $0$. This quantity has a remarkable property: if $P= P'\cup P''$, where $P, P', P''$ are geodesic polygons, then
$$ \delta(P)=\delta(P')+\delta(P'')-\delta(P'\cap P'') $$
which shows that $\delta$ behaves like a finitely additive measure. It can be extended to a countably additive measure on $M$, and as such, it turns out to be absolutely continuous with respect to the the volume measure $dV_g$ defined by the Riemann metric $g$. $\newcommand{\bR}{\mathbb{R}}$ Thus we can find a function $\rho: M\to \bR$ such that
$$\delta= \rho dV_g. $$
More concretely for any $p\in M$ we have
$$\rho(p) =\lim_{P\searrow p} \frac{\delta(P)}{{\rm area}\;(P)}, $$
where the limit is taken over geodesic polygons $P$ that shrink down to the point $p$. In fact
$$ \rho(p) = K(p). $$
Now observe that if we have a geodesic triangulation$\newcommand{\eT}{\mathscr{T}}$ $\eT$ of $M$, the combinatorial Gauss-Bonnet formula reads
$$\sum_{T\in\eT} \delta(T)=2\pi \chi(M). $$
On the other hand
$$\delta(T) =\int_T \rho(p) dV_g(p), $$
and we deduce
$$ \int_M \rho(p) dV_g(g)=\sum_{T\in \eT}\int_T \rho(p) dV_g(p) =\sum_{T\in\eT} \delta(T)=2\pi \chi(M). $$
For more details see these notes for a talk I gave to first year grad students a while back.
As for the second question, perhaps the most general version of Gauss-Bonnet uses the concept of normal cycle introduced by Joseph Fu.
This is a rather tricky and technical subject, which has an intuitive description. Here is roughly the idea.
To each compact and reasonably behaved subset $S\subset \bR^n$ one can associate an $(n-1)$-dimensional current $\newcommand{\bN}{\boldsymbol{N}}$ $\bN^S$ that lives in $\Sigma T\bR^n =$ the unit sphere bundle of the tangent bundle of $\bR^n$. Think of $\bN^S$ as oriented $(n-1)$-dimensional submanifold of $\Sigma T\bR^n$. The term reasonably behaved is quite generous because it includes all of the examples that you can produce in finite time (Cantor-like sets are excluded). For example, any compact, semialgebraic set is reasonably behaved.
How does $\bN^S$ look? For example, if $S$ is a submanifold, then $\bN^S$ is the unit sphere bundle of the normal bundle of $S\hookrightarrow \bR^n$.
If $S$ is a compact domain of $\bR^n$ with $C^2$-boundary, then $\bN^S$, as a subset of $\bR^N\times S^{n-1}$ can be identified with the graph of the Gauss map of $\partial S$, i.e. the map
$$\bR^n\supset \partial S\ni p\mapsto \nu(p)\in S^{n-1}, $$
where $\nu(p)$ denotes the unit-outer-normal to $\partial S$ at $p$.
More generally, for any $ S$, consider the tube of radius $\newcommand{\ve}{{\varepsilon}}$ around $S$
$$S_\ve= \bigl\lbrace x\in\bR^n;\;\; {\rm dist}\;(x, S)\leq \ve\;\bigr\rbrace. $$
For $\ve $ sufficiently small, $S_\ve$ is a compact domain with $C^2$-boundary (here I'm winging it a bit) and we can define $\bN^{S_\ve}$ as before. $\bN^{S_\ve}$ converges in an appropriate way to $\bN^{S}$ as $\ve\to 0$ so that for $\ve$ small, $\bN^{S_\ve}$ is a good approximation for $\bN^S$. Intuitively, $\bN^S$ is the graph of a (possibly non existent) Gauss-map.
If $S$ is a convex polyhedron $\bN^S$ is easy to visualize. In general $\bN^S$ satisfies a remarkable additivity
$$\bN^{S_1\cup S_2}= \bN^{S_1}+\bN^{S_2}-\bN^{S_1\cap S_2}. $$
In particular this leads to quite detailed description for $\bN^S$ for a triangulated space $S$.
Where does the Gauss-Bonnet formula come in? As observed by J. Fu, there are some canonical, $O(n)$-invariant, degree $(n-1)$ differential forms on $\Sigma T\bR^n$, $\omega_0,\dotsc, \omega_{n-1}$ with lots of properties, one being that for any compact reasonable subset $S$
$$\chi(S)=\int_{\bN^S}\omega_0. $$
The last equality contains as special cases the two formulae you included in your question.
I am aware that the last explanations may feel opaque at a first go, so I suggests some easier, friendlier sources.
For the normal cycle of simplicial complexes try these notes. For an exposition of Bernig's elegant approach to normal cycles try these notes.
Even these "friendly" expositions with a minimal amount technicalities could be taxing since they assume familiarity with many concepts.
Last, but not least, you should have a look at these REU notes on these subject. While the normal cycle does not appear, its shadow is all over the place in these beautifully written notes.
Also, see these notes for a minicourse on this topic that I gave a while back.
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
Yes, there's close relationship between angle defect and curvature. It gets called the Bertrand–Diquet–Puiseux theorem:
$$\kappa(p) = \lim_{r\to 0^+} 3\frac{2\pi r-C_p(r)}{\pi r^3}$$
$$\kappa(p) = \lim_{r\to 0^+}12\frac{\pi r^2-A_p(r)}{\pi r^4 } $$
where $C_p(r)$ is the circumference of the circle of radius $r$ centred at $p$ and $A_p(r)$ the area.
These formulas are for smooth surfaces but you can think of the first formula as describing $\kappa$ as an angle defect, and the 2nd formula as describing how $\kappa$ gets concentrated at the vertices for a triangular mesh.