It should be noted that the ring $\mathbb{Z}_p[[T]]$ is the [Iwasawa algebra][1] of the additive group $\mathbb{Z}_p$, and there is a nice structure theory for finitely generated modules over it. See the linked Wikipedia article, and the references therein, as well as e.g. chapters 7 and 13 of Washington's *Introduction to Cyclotomic Fields* or chapter 5 of Lang's *Cyclotomic Fields* (and for a much more general account, Bourbaki's *Commutative Algebra* ch. VII §4). However, this structure theory works "up to pseudo-isomorphism" i.e. up to so-called "pseudo-null" modules which in this case are exactly the ones with finite cardinality. All subquotients which are finite-dimensional $\mathbb{F}_p$-vector spaces are in this class; on the other hand, the rank is invariant under pseudo-isomorphism. So it is highly unlikely that the dimensions in your question will give you any non-trivial information about the rank of your module, as illustrated by abx's answer.

**Edit**: When you consider the $\mathbb{F}_p[[T]]$-module structure as you did between two edits, from this structure theory you get

$rank_{\mathbb{Z}_p[[T]]} M = rank_{\mathbb{F}_p[[T]]} (M/(p)) - rank_{\mathbb{F}_p[[T]]} (M[p])$

but not much more, as far as I can see.



 [1]:http://en.wikipedia.org/wiki/Iwasawa_algebra