Let me first refer you to three references pedagogically treating
Instantons in quantum mechanics: 1)Riccardo Rattazzi's lecture notes treating instantons in nonsupersymmetric quantum mechanics. In these notes the anharmonic oscillator model is elaborated with great detail 2) Philip Argyres lecture notes treating instantons in supersymmetric quantum mechanics. The model considered in these lecture notes is a simplified one dimensional version Witten's model in flat space 3) Salomonson and van-Holten's original article, where they elaborate in detail the same model treated by Argyres. This article can be used to fill the gaps in Argyres' notes.
Regarding the first question:
First, one must notice that if we take a classical solution in which the fermionic coordinates are nonvanishing, then both the bosonic and fermionic coordinates acquire Grassmann components. The bosonic coordinate becomes an even Grassmann number and the fermionic and odd one. The action $S$ itself becomes an even Grassmann number. This casts a difficulty in the interpretation of $e^{-S}$ as a tunneling rate between the degenerate vacuua.
Salomonson and van-Holten (like Witten) discard the fermions (i.e., substitute zero for the fermion's "classical field"). They justify this substitution by the requirement of keeping the action finite (which is a crucial requirement because $e^{-S}$ is proportional to the transition rate between the vacuua). However, Akulov and Duplij find the general solution for the same model with a nonvanishing fermionic coordinate. They find that the Grassmann number contribution to the action identically vanishes and the action is equal to the classical action with the fermions discarded. This partly justifies the discarding of the fermions (Further justification will be given in the fermionic zero mode discussion). In addition Akulov and Duplij find that unlike the action, the Grassmann number dependence does not generally vanish in the instanton topological charge; this contribution vanishes exactly only for potentials breaking supersymmetry which is reminiscent of the vanishing of the Witten's index when supersymmetry is spontaneously broken. Furthermore Akulov and Duplij extend their analysis to a non-supersymmetric model and find that in this case the Grassmann number contribution to the action does not vanish.
The treatment of zero modes:
The fermion and boson determinants excluding the zero modes exactly cancel. As explained in Argyres, zero modes in the fermionic sector cause the partition function to vanish. However, the correction to the ground state energy entails the insertion of a supersymmetry generator (equation 4.16 in Argyres), this insertion is what is necessary to make the fermionic path integral nonvanishing, since for Grassmann variable $\int d\psi_0 = 0$ while $\int \psi_0 d\psi_0 = 1$.
Now, if we adopt the Akulov and Duplij method, the contribution of the classical fermionic field to the path integral vanishes because as already mentioned, the action does not depend on the classical fermionic variables, thus there is no insertion in the classical component, thus its contribution vanishes.
The bosonic zero mode corresponds to the instanton's collective coordinate (moduli space). Geometrically, this coordinate is the central time $t_0$ of the classical kink solution; and the solution satisfies the equations of motion for all $t_0$ values. The correct evaluation of the path integral requires to perform the bosonic integration on the nonzero modes which is finite, then integrate over the moduli space which entails the integration over $t_0$.
The path integral even in this simple case is quite cumbersome and its explicit calculation is given in Salomonson and van-Holten.
For a rigorous and detailed computation of the instanton path integral for the
Witten's model, please see Alice Rogers article.
First, the reasoning in the question about isomorphism classes of bundles is wrong, because the $\check{H}^1(M,G)$ from the linked math.SE post is not the cohomology of $M$ with coefficients in $G$, but actually the Čech cohomology of $M$ for the sheaf $\mathscr{G} : U\mapsto C^\infty(U,G)$.
However, this indeed has a relation to the cohomology of $M$ itself for $G = \mathrm{U}(1)$, via
$$ 0 \to \mathbb{Z} \to \mathbb{R} \to \mathrm{U}(1) \to 0$$
which turns into
$$ 0 \to C^\infty(U,\mathbb{Z})\to C^\infty(U,\mathbb{R}) \to C^\infty(U,\mathrm{U}(1))\to 0$$
since $C^\infty(M,-)$ is left exact and one may convince oneself that this particular sequence is still exact since the map $C^\infty(M,\mathbb{R})\to C^\infty(M,\mathrm{U}(1))$ works by just dividing $\mathbb{Z}$ out of $\mathbb{R}$. Considering this as a sheaf sequence $0\to \mathscr{Z}\to \mathscr{R} \to \mathscr{G} \to 0$, $\mathscr{Z} = \underline{\mathbb{Z}}$ for $\underline{\mathbb{Z}}$ the locally constant sheaf since $\mathbb{Z}$ is discrete, and the sheaf of smooth real-valued functions on a manifold is acyclic due to existence of partitions of unity, so taking the sheaf cohomology one gets
$$ \dots \to 0 \to H^1(M,\mathscr{G}) \to H^2(M,\underline{\mathbb{Z}})\to 0 \to \dots$$
and thus $H^1(M,\mathscr{G}) = H^2(M,\underline{\mathbb{Z}}) = H^2(M,\mathbb{Z})$ where the last object is just the usual integral cohomology of $M$. Hence, $\mathrm{U}(1)$ bundles are indeed classified fully by their first Chern class which is physically the (magnetic!) flux through closed 2-cycles, and the existence of non-trivial $\mathrm{U}(1)$-bundles would imply non-trivial second cohomology of spacetime (or rather of one-point compactified spacetime $S^4$ since one should be able to talk about the field configuration "at infinty" and the bundle being framed at infinity). Indeed, since $H^2(S^4) = 0$, the existence of $\mathrm{U}(1)$-instantons would contradict the idea that spacetime is $\mathbb{R}^4$.
For general compact, connected $G$, it turns out the possible instantons are pretty much independent of the topology of $M$ because a generic instanton is localized around a point, as the BPST instanton construction shows - the instanton has a center, and one may indeed imagine the Chern-Simons form to be a "current" that flows out of that point, giving rise to a nontrivial $\int F\wedge F$.
Topologically, one may understand this by imagining $S^4$, and giving a bundle by giving the gauge fields on the two hemisphere, gluing by specifying a gauge transformation on the overlap of the two, which can be shrunk to $S^3$, i.e. the bundle is given by a map $S^3\to G$, and the homotopy classes of such maps are the third homotopy group $\pi_3(G)$, which is $\mathbb{Z}$ for semi-simple compact $G$. Since the "equator" can be freely moved around the $S^4$, or even shrunk arbitrarily close to a point, this construction does not in fact depend of the global properties of $S^4$, it can be done "around a point".
Thus, instantons in general do not tell us anything about the topology of spacetime.
This answer has been guided by the PhysicsOverflow answer to the same question.
Best Answer
Yes, an instanton is a classical solution to the Euclidean equations of motion with finite action. Its topological charge is given by $k = \frac{1}{8\pi}\int \mathrm{tr}(F\wedge F)$ which is the integral of the divergence of the Chern-Simons current.
There are many different instantons possible. A generic instanton for $\mathrm{SU}(2)$ and topological charge 1 is given by the BPST instanton $$ A_\mu^a(x) = \frac{2}{g}\frac{\eta_{\mu\nu}^a(x-x_0)^\nu}{(x -x_0)^2 - \rho ^2}$$ where $x_0$ is the "center" of the instanton and $\rho$ its scale, also called the radius. The $\eta$ is the 't Hooft symbol.
A large class of instantons of topological charge $k$ may be described as follows: Transforming the BPST instanton by the singular transformation $x^\mu \mapsto \frac{x^\mu}{x^2}$ leads to the expression $$ A_\mu^a(x) = -\eta_{\mu\nu}^a \partial^\nu\left(\ln\left(1+\frac{\rho^2}{(x-x_0)^2}\right)\right)$$ for the transformed instanton, and one now makes the more general ansatz $$ A_\mu^a(x) = -\eta_{\mu\nu}^a \partial^\nu\left(\ln\left(1+\sum_{l=1}^k \frac{\rho_l^2}{(x-x_{0,l})^2}\right)\right)$$ which leads to an instanton solution of topological charge $k$. This construction can be generalized to other non-Abelian gauge groups.
The generic construction of all instantons on four-dimensional spacetimes of gauge group $\mathrm{SU}(N)$ is given by the ADHM instanton, see also the original paper "Construction of instantons" by Atiyah, Drinfeld, Hitchin and Manin.