Say I have $X\_{ij}$, $j \le i$ with the property that $X\_{ij}$ are centered and identically distributed and $E(X\_{ij} X\_{ij'}) = o(\exp(-i)))$. Then does $\sum\_j X\_{ij}$ have Gaussian domain of attraction? 


As a related question, if $X\_1, X\_2, X\_3$ are identically distributed and centered and $E(X\_i X\_j) = c$, what bound can I get for $E(X\_1 X\_2 X\_3)$ in terms of $c$?



Does central limit theorem hold for general weakly dependent variables?
Not necessarily. One has to impose more restrictive mixing and moment conditions. A classical book is:


Ibragimov I.A., Linnik Yu.V. *Independent and stationary sequences of random variables*


There is a long-standing question asked by Ibragimov: is $\phi$-mixing and finiteness of second moment sufficient for CLT to hold for a stationary sequence?


Also, there are various concepts of dependence. For example, if your r.v.'s are associated (i.e. satisfy FKG inequalities) and the covariance decays as you describe, then CLT holds. 


UPD. As for the second part of your question: you cannot estimate higher-order moments in terms of lower-order ones unless the joint distributions have some special structure.


