@Marc You are right; I misread what they meant by $P$. The key point is anyway that $\psi$ isn’t zero at the south pole (as it would be for the bubbles you are worried about). I’ve modified accordingly, and hopefully not made any more mistakes guessing at their notation.

Your explanation for the degeneracy of the metric is correct.

The operator $u \mapsto \sigma_k(\lambda(A_{g_u}))$ is not linear, even after taking a power to make it homogeneous of degree $1$. For this reason, you do not have a linear representation formula.

For your second question: I computed the linearization $D_v\mathcal{F}_\mu(w)\rvert_{w=1}$. To do this, you just need to pick an arbitrary path $w_t$ with $w_0=1$ and $\dot w=v$. Mine is the straight line path with respect to your parameterization $g_w = w^{\frac{4}{n-2}}g$ of the conformal metric. For the first question, the point is that $\varphi_t$ is a conformal diffeomorphism. Its role is similar to the role of the Kelvin transform in the linked question.

In general you don’t have an equation which makes sense at $z=0$. There are some removable singularity theorems under appropriate assumptions on $u$ and $K$. You can probably find details in papers by Yanyan Li or Maria del Mar Gonzalez from the early- to mid-2000s, but I don’t know the most general reference.

Right, $g_0$ is the flat metric. Without knowing your background and interests, I'm unsure of what would be a "friendly" reference. A good starting point might be one of Alice Chang's survey articles on fully nonlinear elliptic equations.

$v_i^j$ denotes the components of the matrix $g_0^{-1}D^2v$, where $D^2v$ is the Hessian of $v$ with respect to the metric $g_0$.

The tensor $g^{-1}A_g$ is a section of $T^\ast M \otimes TM \cong \mathrm{End}(TM)$, so for each point $p \in M$, the value of $g^{-1}A_g$ is a linear map, and hence one can meaningfully talk about the eigenvalues (or think of it as an $n \times n$ matrix). Without a metric, there is no way to talk about the eigenvalues of a section of $T^\ast M \otimes T^\ast M$. This is what I meant by needing to raise an index to define $\sigma_k$.

To go from an elliptic complex, as usually defined, to the Hodge decomposition theorem, you need to pick metrics on the vector bundles. If you only pick a pseudo-Riemannian metric, as in my first example, then you don’t get the decomposition. In my second and third examples, you do not (need to) pick a Riemannian metric to get the decomposition, but rather just a sub-Riemannian metric (the Levi form). Given your question about harmonic sections, it is better to think of an elliptic complex as requiring also the specification of a metric on each of the vector bundles.