Let me also mention the book "Coxeter matroids" by Borovik, Gelfand, and White. Note that the definition they consider is analogous to (and in Grassmannians, reproduces) that of matroid, not realizable matroid, which is more in line with what you're asking about.

He's got a video on this result up, at youtube.com/watch?v=6pzLJllRyoY

Yes, dot for conjugation.

I think it's interesting to note that Killing counted $F_4$ twice, without realizing it.

Incidentally, your subgroup is a Levi subgroup, so branching can be computed using crystals. That would take some work and not get you results as precise as what's already mentioned.

Somehow this reminds me of $n+m \choose m$, which is a polynomial for $n$ or $m$ fixed but isn't a polynomial.