In Riehl's book Categorical Homotopy Theory, the dual case (colimits vs homotopy colimits for sequences of cofibrations) is treated in Example 11.5.11 in the context of cofibrantly generated simplicial model categories. She shows in that case that it is actually a projectively cofibrant diagram. I think that in your case her argument can be adapted to show it is an injectively fibrant diagram (if the injective model structure on such diagrams exists). So again limit will compute the homotopy limit.

I might be mistaken, but I think it can be proven as follows. Isn't the index category of your limit a Reedy category (with only one kind of map)? Then if the maps are fibrations, then it is a Reedy fibrant diagram. Thus in that case the limit also computes the homotopy limit.

This is part of the general yoga of the Dold-Kan correspondence between cosimplicial abelian groups and cochain complexes. There are two ways to go from a cosimplicial abelian group to cochains. You either use the Moore complex (whose differential is the alternating sum of the face maps) or using the normalized Moore complex. The latter is an equivalence of categories, but both complexes have the same cohomology. A reference for the chain/simplicial ab-group case can be read here: ncatlab.org/nlab/show/Moore+complex see thm 3.3.

Yes, that is what I mean by unitors.

If you allow non-trivial unitors as well, then things are more complicated. It is true that then any cocycle can be used. The 3-cocyclce almost determines the unitors. The associator and unitors are determined by the 3-cocycle plus the value $\rho_1 = \lambda_1$ of the unitor on the unit object (which can be non-trivial in general). Then you have to consider these up to monoidal isomorphism, and the first step might be to show that without changing the cocycle you can always reduce to the case that $\rho_1 = \lambda_1$ is trivial.

I agree with you that if your unitors are trivial, then you need to use normalized cocycles. Fortunately normalized cocycles compute the same group cohomology as unnormalized cocycles. There are many non-trivial examples. Degree 3-cohomology corresponds to "crossed modules", which you can google to get more information and examples. Probably the easiest non-trivial example is when $G=\mathbb{Z}/n$. Then $H^3(G; A)$ is the n-torsion subgroup of $A$ (here we take A to have trivial $G$ action). Each non-trivial element is represented by a non-trivial normalized 3-cocycle.

This is not too hard to see from the point of view of the Pontryagin-Thom construction. You adjoin the map into a map $S^3 \to S^2$, and then look at the inverse image of a generic point (not the base point or the fixed points of the rotation). You also keep track of the framing from $\mathbb{R}^2$ at that point. The inverse image is a framed unknot whose framing "twists" around once, and so has self linking number (i.e. Hopf invariant) equal to one.

Doesn't the usual proof work? Any compact subspace of $E$ lies in some $E_\xi$ where $\xi < \lambda$ (but not necessarily a limit ordinal). So $\pi_i(E, p^{-1}(y_0))$ is the direct limit of the $\pi_i( E_\xi, p^{-1}(y_0))$, and so $\pi_i (E, p^{-1}(y_0)) \to \pi_i(Y, b)$ is an isomorphism if each map $\pi_i(E_\xi, p^{-1}(y_0)) \to \pi_i (X_\xi, y_0)$ is an isomorphism. This is the same argument as given in Hatcher Lemma 4K.3.