Reddit mentions of Functional Syntax and Universal Grammar (Cambridge Studies in Linguistics)

This may or may not be a helpful answer to you, but I find that possibly both of your questions might be helped if you take a look at Role and Reference Grammar. It's an attempt (that works mostly pretty well IMO) to make an integrated syntax and semantics framework that is very explicitly not based on Indo-European prototypes - a lot of it is based on the mechanics of languages like Lakhota, Dyirbal and Tagalog. RRG should give you both a good framework to talk about case relations and an expansion of your understanding of what syntax should look like. (I don't at all like the notation RRG uses for semantics and case relations, but the concepts are good.)

It's not necessarily an easy topic to read up on, though; the guy who came up with it doesn't always have the clearest writing style. The class I took on it used this book; I read this book first, though, and thought it somewhat unintentionally functioned as a pretty decent introduction (ignore its title, though; and beware that some aspects of RRG have changed since it was written).