If I have a simplicial commutative monoid, $A_\bullet$, let $B_n$ be the intersection in $A_n$ of the kernels of all the $d_i$ except $d_n$, and let the differential be the last map $d_n$. Doesn't this define a chain complex $B_*$?

that is a nice argument!

Take any non-semisimple Frobenius algebra A (in the classical sense, i.e. in vector spaces). Then $A \otimes 1$ is a non-semisimple Frobenius algebra in your tensor category (that is, tensor A with the unit).

@PeterLeFanuLumsdaine That is what I thought, but now I am not sure. I read Moishe's response to my comment as saying that he isn't sure Vogt's claim there is justified. Vogt cites a paper of Epstein "Foliations with all leaves compact". If I am reading Epstein's paper correctly he shows an equivalence of "bounded volume-of-leaf" to several other conditions but this only includes a Seifert fibration if the foliation is further assumed to be C^1. Maybe this condition can be removed or relaxed? If not it looks like there is still room for a positive answer in the non-differentiable C^0 case.

These foliations do not satisfy the continuity condition in the OP. In the introduction to the first of Vogt's papers mentioned he notes "A compact foliation has a locally bounded volume-of-leaf function if and only if it is a Seifert fibration". In particular the foliation he constructs does not have a locally bounded volume-of-leaf function. The "close images" criterion for continuity surely implies a locally bounded volume-of-leaf function.

It is much harder than I thought it would be to come up with a counterexample!

Graeme Segal. Cohomology of topological groups. In Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69), pages 377–387. Academic Press, London, 1970.

If memory serves, this is described in Milnor and Stasheff's book Characteristic Classes... yes it looks like it is Theorem 19.7, attributed to Wu.

This is a good question. As your example shows, the map $\rho_V^U$ need not be an isomorphisms. That seems to be a mistake. I think it is still true that $\mathcal{F}$ is a locally constant presheaf, but this might depend on the definition that Bott and Tu are using. Is it enough for the sheafification to be locally constant?

Related to this, but in a different direction related to the second half of Tim's comment, is the paper, "QUILLEN THEOREMS Bn FOR HOMOTOPY PULLBACKS OF (∞,k)-CATEGORIES", by Barwick and Kan which discusses this issue in the model of relative categories among others.

(cont) The homotopy coherent nerve is a right quillen equivalence between this model structure and the Joyal model structure on simplicial sets. So, you can use this to give some sort of answer to part B too. For example if F is a fibration in the Bergner model structure, then I think the homotopy fiber is just the nerve of the fiber. I am not sure if that is what you want.

The usual Quillen theorems A/B concern properties of a functor of categories and deduce consequences about the corresponding map of nerves, regarded in the standard Quillen-Kan model structure (i.e. regarded as spaces). You seem to be asking instead about consequences of (homotopy coherent) nerves regarded as quasicategories/infty-categories. If that is the case, then there is a precise answer to A - these F are the Dwyer-Kan equivalences. These are the weak equivalences in the Bergner model structure on simplicially enriched categories. (cont)