EFA with "found" data The textbook procedure for EFA is to start by designing the data collection, then you run the EFA to locate (we hope) the latent variables, therein.
Does that mean it is not ever valid to conduct and EFA with "found" data, such as a public data source?
If it is valid, does it mean I should insert a pre-EFA step of filtering the data according to a theoretical model or should I "take it as it is" and let the chips fall where they may?
In this particular case, I do have a theoretical model, and it is likely that, regardless of the strengths of correlations, some of the variables would have no connection by theory to the model.

Exploratory Factor Analysis (EFA) is often used with "found" data, and there is no mathematical, theoretical, or statistical reason why it couldn't be.
The logit of EFA is simply that responses to a set of observed variables are actually due to some smaller number of "latent" variable (plus item specific error). So if you saw that someone else did a survey of people in an organization and asked a ton of questions about how people felt about their boss, you could theorize that responses to lots of those questions are heavily determined by a latent variable: "thinking your boss is good," and you could use EFA to see if that's the case, even if the person who designed that survey had never heard of factor analysis. EFA might find that all of the items load on a single factor, or that each question is basically it's own construct and that there is no underlying latent factor at all. That's the beauty of EFA: it doesn't make any assumptions about what you will find.
This is contrast to CONFIRMATORY factor analysis (CFA). CFA has the same basic assumption as EFA (observed responses are due to some number of latent factors plus item specific error) but in CFA  you already assume that you know exactly how many latent factors there are, and just want to see how strongly each relate to the specific items in that particular dataset.  You therefore probably wouldn't want to use CFA on a set of items that were not designed to measure a specific set of latent factors.
