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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 V_κ 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 V_γ is elementary in V_κ, and so these also will be models of ZFC. It is an exercise to show that the least γ for which V_γ 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 smaller set (since 2^ω would be already too large) and it is also regular. But we never intend that ω₁ is an inaccessible cardinal.

Your remark that V_κ is closed under pairs when κ is inaccessible actually doesn't need any amount of inaccessibility. If x and y are sets in any V_α, 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, for infinite rank sets), and so every V_λ is closed under pairing for any limit ordinal λ.

Finally, yes, if V_λ is closed under pairing, then 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(V_α^M).

There is a famous result of Laver concerning the left distributive algebra of nontrivial elementary embeddings j:V_λ to V_λ. The first results characterizing normal forms in the free algebra with one generator used such embeddings, with the accompanying very large large cardinal hypothesis. For example, Laver produced a decision procedure, which was only known to work under these enormous large cardinal assumptions. Later, the large cardinal hypotheses were removed and the algebra became studied apart from the large cardinals, but the basic properties were definitely inspired and discovered by knowledge of what the large caridnals were like. The basic operation in this algebra is known as application, and is exactly the operation that you mention.