@IgorBelegradek The base space is $S^9$, so the only possible Stiefel-Whitney or Pontryagin class which could possibly be non-zero is $w_9$. This too vanishes, as the OP explained.

You can do this using John Pardon's version of dagger category, but where you modify it slightly. Instead of the category $Fun(\Delta^1, \mathcal{C})_{isom}$, you use a category of arrows where the morphisms are squares with sides coming from your distinguished "subcategory" of "unitary isomorphisms". Then the problem which I pointed out goes away since then you only need naturality with respect to the unitary isomorphisms, which is fine.

Interesting question! Modular tensor categories correspond to partially extended tqfts, assigning data to 1,2, and 3-manifolds. Non-extended 3D tqfts are much more complicated (see arxiv.org/abs/1509.06811 and arxiv.org/abs/1408.0668). Now if we are just considering partially extended TQFTs (assigning data to d, d-1, and d-2 manifolds) then there are no such examples in dimension four, and I think I can show that there are no such examples for all dimensions d. The general case is a little more involved/original than a typical MO answer. Maybe I'll write it up as a note?

(2) Weak equivalences are supposed to be inverted, but sometimes when you invert some arrows it forces other arrows to be inverted. Saturation is important since it tells you you know exactly which arrows get inverted. (1), 2-out-of-6 is something you can actually check sometimes. Saturated homotopical categories can still be a mess to deal with. It is nice to have the definition, but I think the more useful concept, at least if you ever want to compute anything, is what Barwick and Kan call a "partial model category".

Also looking into it a little more carefully it seems like it won't be possible for this manifold to have an honest Spin(7)-structure (as opposed to an almost Spin(7)-structure). The manifold I want does not seem to satisfy the conditions of Thm 10.6.1 in Joyce's "Compact Manifolds with Special Holonomy". I think that means I was being foolish to think that this manifold could come from these more established geometries.

The tangential structure that I really want is that $\tau_M \cong f^*(\gamma_\mathbb{O})$ for some map $f: M \to \mathbb{OP}^1$. This gives you, in particular, seven different almost complex structures, any two of which generate a quaternion algebra. You can get this by starting with other "larger" tangential structures such as quaternionic-Kahler or Spin(7), but then you will have to require that more characteristic classes vanish.

If the source and target maps $s,t: C_1 \to C_0$ are Hurewicz fibrations, then this is a very easy corollary of the main result of R. W. Kieboom, A pullback theorem for cofibrations (1987).