The space of n-flats in $R^m$ is sometimes called an affine Grassmannian. Wikipedia gives two different definitions for this term, one of which seems to be what you want. The affine Grassmannian in this sense contains the ordinary Grassmannian and is a vector bundle over it, so it is a manifold but it is not homeomorphic to a euclidean space since it has the same homology and homotopy groups as the ordinary Grassmannian, and many of these groups are nontrivial, starting with the fundamental group.

A textbook reference for this argument is Example 4L.7 on page 495 of my Algebraic Topology book.

One can construct $OP^2$ by attaching a 16-cell to $S^8$ via the Hopf bundle map $S^{15}\to S^8$, and this Hopf bundle can be constructed just using the octonions. (Details can be found in Example 4.47 of my algebraic topology book.) So Lie groups aren't needed to construct $OP^2$. If one had a bundle $S^7 \to S^{23} \to OP^2$, couldn't one attach a 24-cell to $OP^2$ via the map $S^{23} \to OP^2$ to construct an $OP^3$ (which doesn't exist)?

And in fact $x$ and $y$ are a basis for the free group they generate, since finitely generated free groups are Hopfian.

In dimension 2 the result is quite a bit older than the theorem of Earle and Eells, which determines the full homotopy type of the diffeomorphism group of a closed orientable surface, not just the mapping class group. The mapping class group of the 2-dimensional torus was known to people like Nielsen, Dehn, and Baer around 1930 or even before. A modern textbook reference is the upcoming book on mapping class groups by Farb and Margalit.

That's a very nice simple inductive formula. It could be fun to work out whether it can be translated into a reasonable closed form answer.

In addition to the original sources, a textbook exposition of the abelian group case can be found in Proposition 3F.12 of my algebraic topology book. (Don't you just hate it when authors are constantly referring to their own books?)

Another reference for the calculation of the complex K-theory of real projective spaces is Atiyah's K-theory book, Proposition 2.7.7.

A decade after Schubert, Waldhausen also came close to the JSJ decomposition, in his work on graph manifolds for example. It almost seems as if Seifert back in the 1930s could have discovered the JSJ decomposition.

That's right, I should have mentioned the condition of being primitively generated. I also don't know of topological examples of more complicated sorts of duality relationships (but I'm not an expert in this area).