Unfortunately, your formulations of the definition of strongly inaccessible are not quite correct. The correct definition is that κ is strongly inaccessible (also known as just plain inaccessible), if κ is uncountable, and whenever β < κ, then the power set of β also has size less than κ, and also κ is regular, which means that it is not the supremum of any set of size less than κ.
This is the not the same as asserting that Vκ being a model of ZFC. (Although, to be sure, this false assertion has appeared surprisingly often in print and I have even heard a famous proof theorist make this assertion to a very large audience of hundreds of logicians.) The reason is that if κ is strongly inaccessible, then there will always be many γ < κ for which Vγ is elementary in Vκ, and so these also will be models of ZFC. It is an easy exercise to show that the least γ for which Vγ is a model of ZFC has cofinality ω, and so is definitely not inaccessible.
This is also not the same as saying that κ is regular and not the size of a power set of a smaller set. The reason is that if, say, CH failed, then ω1 would not be the size of the power set of any set (since 2ω would be too large) and it is also regular. But we never intend that ω1 is an inaccessible cardinal.
Your remark that Vκ is close under pairs when κ is inaccessible actually doesn't need anything more than κ being a limit ordinal. If x and y are sets in any Vα, then the pair (x,y) appears just a few steps later (and actually, one can use flat pairing function that do not increase rank at all), and so every Vλ is closed under pairing for any limit ordinal λ.
Finally, yes, you can apply such functions to themselves, and this idea is used quite often when we have elementary embeddings defined on models of ZFC. For example, if j:V to M, then j(j) is a function defined on M, into some structure j(M), which will be the union of j(VαM).