(2) seems suspiciously straightforward. In some sense, adding $r\subset {\mathbb N}$ as a predicate you are adding something "very similar" to $\mathbb N$ or $\mathbb Z$ (but then (3) seems strange - what kind of generic extension gives (3)?) --- Now, for the Question: The set C of all Cohen reals is comeager in the new reals. My quick guess would be that some careful use of genericity implies non-definability of $\mathbb Z$.

Thanks to Assaf for pointing out Andrzej's paper, and to Andrés for linking this on twitter!

In many senses, the algebraist you mention is correct: QE for many theories is useful, but not quite a model theoretic fact (at least not in the same way as many of the other deeper model theoretic facts used in the constructions mentioned by Goodrick and others below). The details of QE proofs draw completely on the algebraic situation you're dealing with, and (mostly) depend on quite basic, quite soft model theory. Now, the examples mentioned by Medvedev (alg. dynamics), Goodrick and others usually depend on geometric model theory, not reducible to algebra, really.

But does the question still make sense for more general ultra-inverse limits (on large index sets, etc.)?

I didn't know about the connection between these questions and the strong independence property! Is this "recent" (well, mod the fact you wrote this answer two years ago) or is this part of the Vive la différance I-III series?