All FOL models/theories share a common 'framework', this being FOL of course. That's why Henkin constructions, skolem functions, etc. work so well with no assumptions of the theory being modeled. However, most of the time the proofs of theorems use only one fact of FOL, that the cardinality of its sentences being the same as the language of the structure. In proofs, the are ordered them in some special way, and then the apply some procedure to each of the sentences and the end result is what was needed (using some properties of the deduction system too).
My (rather vague and open ended) question is, is there any better intuition/ordering of FOL sentences? For example, in Chang & Keisler's Model Theory they use the algebraic properties of boolean algebra and argue using properties of it's ultrafilter structure. Are there any other, easier-to-wrap-your-head-around, constructions? Is there any geometric interpretation of skolem functions? Elementary equivalent structures?