It should be straightforward to calculate the index of the linearized Cauchy-Riemann operator on punctured Riemann surface with some properly chosen weighted Sobolev norm. I guess your formula should be correct.

You can consider the 1-dimensional case, when symplectic form coincides with the volume form. The thing you obtained by integrating along the fibre is a section of the dual bundle of the density bundle of the base.

One should be able to give an argument by looking at the normal form of the singularity, i.e., f = -x^2 + y^2 + a and the metric is Euclidean.

Yes, I see what the problem is. There could be very "thin" solutions.

Thanks. But if we consider moment maps which are like "quadratic functions", for general compact Lie group, is there still counter-examples? Or is there some other conditions to guarantee the similar situation in the compact case (i.e., the convergence of gradient flow)?

Actually recently Jake Solomon defined certain functional (of Calabi type) on a Hamiltonian deformation class of Lagrangians. The minima of this functional should be special Lagrangians and they should be the mirror of the Hermitian-Einstein metrics on a stable vector bundle.

If $du$ is defined almost everywhere, then $J \circ du$ makes sense wherever $du$ has a value. It doesn't refer to any embedding.