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I've thought about similar things recently. Perhaps this should really be a linear algebra question first! Take $X$ to be a point, so you are simply working with a complex vector space $F$ endowed with a real structure, i.e., a descent datum isomorphism from $F$ to its conjugate space. You want to classify Hermitian forms on $F$ which are compatible, in a sense, with the descent datum. If this is what I think it is, the answer should be something like $GL(F_R) / O(F_R, B)$, where $F_R$ is the conjugation-invariant subspace, and $O(F_R, B)$ is a certain orthogonal group.
The parabolic subgroups of $F_4$ can be identified with stabilizers of certain "flags" in $J_0$ (the trace-zero elements in the 27-dimensional Jordan algebra) -- this is the "metasymplectic geometry" which I think was described by Freudenthal in the 1950s. There are four maximal parabolics (up to conjugacy) in $F_4$. One such parabolic can be viewed as the stabilizer of a norm-zero line (for the cubic norm) in $J_0$. Other maximal parabolics are stabilizers of certain "lines", "planes", and "symplecta" in the projective space $PJ_0$. It's hard to find references! Try Freudenthal or Tits.
One more - yes, when I wrote $\psi = (v_\lambda^{(1)}, \ldots, v_\lambda^{(m_\lambda)} )_{\lambda \in \hat G}$, I mean that $\psi$ is the concatenation of such tuples as you suggest, where $\lambda$ varies over the set of irreducible representations of $\hat G$ (one for each isomorphism class of such irreducible representations).
@Arnirbit: The representation of $G$ on $Ind_H^G W$ is by left-translation of functions, i.e., $[g f](x) = f(g^{-1} x)$. Often it is not irreducible, but any irreducible summand of $Ind_H^G W$ will still consist of some functions from $G$ to $W$, stable under left-translation by $G$. For more on the Casimir element, you might consult a good book on Lie groups -- perhaps Varadarajan's book might appeal to you.
@José: Thanks for the note on the typo - it's fixed now. @Anirbit: Here $W$ is a representation of $H$, and $Ind_H^G W$ is a representaion of $G$. It consists of certain functions from $G$ to $W$. These functions are not quite $H$-invariant, but instead are "$(H,\tau)$-invariant" in the way described by the identity mentioned. Such functions are in natural bijection with sections of a vector bundle as you wish. In fact, this can be used to define the vector bundle on $G/H$ associated to a representation $W$ of $H$.
Ooof! I don't know of anyone who would want to write down a 27 by 27 matrix, one for each of the 52 basis elements of the Lie algebra of type F_4! I have worked with this representation in the past, but always saving space by working with 3x3 octonionic Hermitian matrices, and certain subgroups of F_4 which act nicely (with easy-to-write formulas).
I don't think there's an analogy between homotopy groups of spheres and the distribution of primes in any way. Yes, they both yield some numbers which are not very well understood -- but you can't say that any two poorly understood sequences are analogous, just because they're both poorly understood! So I think these questions are not that well-motivated, at least by the middle paragraph above. On the other hand, I too wonder about questions like "what are homotopy groups of spheres good for, outside of the easy and obviously useful cases?"
