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For a general fibration the argument is almost identical, with one additional wrinkle. When you restrict the fibration to the n-cell, it is not necessarily a product $D^n \times F$. But it is homotopy equivalent to such a product. So now we don't have an isomorphism but just a homotopy equivalence $p^{-1}(B/B') \simeq (D^n /S^{n-1}) \times F$. In cohomology though, we still get an isomorphism, so the argument precedes as before.
Only the CW structure on the base B is used in the argument. (Plus an assumption that the spaces E, E' etc are sufficiently nice to switch from relative cohomology to reduced cohomology of the quotient, and to apply excision, and that F has a Euler characteristic). The CW structure of F and/or E is not used.
The only way I know of to prove that $Bord_n^{(-)}$ preserves homotopy colimits uses the cobordism hypothesis. You don't need the full version, you can prove it more or less directly using induction and Thm 3.1.8 in Lurie's paper (the "inductive" version of the cob. hypo.). It is interesting to note that this fact is only true in the fully-local case, where your bordism category is extended all the way down to points. The corresponding statement for the partially extend bordism higher category is actually false.
I would say that there is an "obvious resemblance" between model categories and weak factorization systems, while what you describe, the (epi,mono) system, is an orthogonal factorization system. Orthogonal factorization systems are very special. For weak factorization systems (mono, epi) on set is a perfectly good example with generic features. See ncatlab.org/nlab/show/weak+factorization+system
You are correct. Lemma 3.21 is actually false. Here is an easy counter example: C is the tensor category Vect[G] of G-graded vector spaces for the finite group G. The forgetful functor to Vect, which forgets the grading, is a tensor functor. This makes Vect a C-module category. We let M = Vect as a right C-module category and N=Vect as a left C-module category. Then clearly $M \boxtimes N \cong Vect$, but a nice exercise shows that $$Fun_{Vect[G]}(M^{op}, N) \cong Rep(G) $$ is the category of $G$-representations. So the functor "I" in lem 3.21 is definitely NOT an equivalence in general.
If I understand the results of that paper correctly, the author only constructs counter examples as lax TQFTs (in the sense of lax symmetric monoidal functor), and not TQFTs in the usual sense.
Note also that if BG satisfies the first definition, then $id \in [BG,BG]$ corresponds to the bundle $EG \in k_G(BG)$. So you have the bundle in the first definition too.
The next theorem (thm 14.4.2) shows that there exists a bundle EG over a space BG satisfying the second defn. Hence by thm 14.1, this BG is also universal in the sense of the first definition. By the Yoneda lemma, any numerable spaces BG and BG' satisfying the first defn are homotopic. Thus if you have a space BG satisfy the first defn, then it is homotopic to the BG constructed in Thm 14.4.2, and hence also satisfies the second defn.
How are you not satisfied with the discussion in Section 14.4 of Tom Dieck's Algebraic Topology book? This explains in detail exactly the relationship between these two perspectives. (Note that in Husemoller's book $k_G(-)$ is isomorphism classes of numerable $G$-bundles).
