Communication allowed in the NC^i model?
The model of parallel computing used in the definition of Nick's class is PRAM (see this wiki page, note: particular policy for handling read/write conflicts is irrelevant for Nick's class). In this model, the processors share memory and a common clock, but are otherwise independent. Therefore, any communication that happens between processors uses memory as the channel. The result of the computation can be announced in some pre specified register.

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Note there are a couple ambiguities in your statements: to show that a problem belongs to $\mathrm{NC}^i$ you can choose $k$ (the power in your polynomial number of processors) depending on the particular problem, while $i$ is fixed. Also, all the quantities should only be specified up to a constant, e.g. $O(\log^i n)$ time and $O(n^k)$ processors.