Definition of topological string theory correlators

Question

About the definition (say in the A-model) $$F_g = \int_{M_g} \langle \prod_{i=1}^{3g-3}\left| \int_{\Sigma_g} G^-_{zz} (\mu_i)^z_{~\bar z}\right|^2 \rangle$$ I have two questions:

1. how are the CFT correlators precisely defined?
2. the $3g-3$ comes from dimension counting: it is equivalent to study perturbations of complex structure $J \mapsto J+\epsilon$, to find the conditions on Beltrami differentials; why in doing that one has to impose vanishing of Nijenhuis tensor? (isn't this automatic in 2d?)

References:

Neitzke and Vafa lectures http://arxiv.org/abs/hep-th/0410178 (correlator definition)

Collinucci lectures http://www.ulb.ac.be/sciences/ptm/pmif/Rencontres/topstrings.pdf (dimension counting)