Correct me if I am wrong but I believe at least conceptually (maybe even rigorously) data of a 1-dimensional TQFT and of a vector bundle with connection are equivalent.

Going into more detail (and consequently making more and more mistakes), a vector bundle with connection allows to assign to a point the fibre over that point, and to a path the monodromy (or holonomy? Yes I am that ignorant) along this path.

Three questions, but closely related:

Does there exist equally layman-ish description of going back from a TQFT to a vector bundle?

Clearly this requires the base to be at least a smooth manifold. Is there known any TQFT-like object (and maybe also connection-like object) that would work in the non-smooth context? Say, for topological manifolds, or even arbitrary finite CW-complexes?

What if one leaves manifolds alone but removes the connection? Is there known a version of TQFT that would work for vector bundles with arbitrarily severe restrictions (say, very-very nice algebraic vector bundles on very-very good algebraic varieties) but without any additional structure?

Two remarks:

Sort of minimal version of the question is whether a vector bundle $p:E\to[0,1]$ comes with any kind of map (relation? correspondence?) between $p^{-1}(0)$ and $p^{-1}(1)$. Vague association with motives comes to mind but that's all my mind offers.

Obviously I am tempted to ask the same about 2D-TQFT. But I am (almost) successfully resisting this temptation.