I'm guessing the original post is not talking about HMDA reportable applications (or it would pretty obviously be the Demographic info used for all of this). I don't know what FDIC does, but I can share what our examiner has told us.
As far as joint apps, I believe our examiners (FRB) say if any borrower falls into race or ethnicity minority, then they would be a part of the protected class. For gender, a joint application is excluded from analysis (just look individual male v. individual female). You'll need to check with your own regulator though. According to our fair lending vendor, none of this is very consistent.
As far a proxy, the Fed put this out a while ago (https://www.consumercomplianceoutlook.org/outlook-live/2013/indirect-auto-lending/
). See the documents at the bottom including the step by step proxy methodology. It's about 6 years old, but our examiners pointed it out to us pretty recently as still being used though a new list of Hispanic names was recently released (I can't find the link). It's technically specific to indirect auto, but they suggested this is more broadly used for non-HMDA consumer loans. It is a very basic proxy, and I know that the CFPB at least uses much more sophisticated BISG proxy analysis. As far as race, geocoding the address and looking at those in minority majority census tracts is used as a proxy for racial minorities can be used.
Bottom line is to ask your examiner what to use. There is not a ton of consistency. Ours has been very responsive. They generally want you to do the correct analysis, and that might be different from examiner to examiner and bank to bank. You also might want to look at it from different angles. If a disparity shows up, then you've got risk.