For an additive $A$ and any morphism $f:X\to Y$ in $C(A)$ one has the following distinguished triangle in the homotopy category $K(A)$: $X\to Y\to Cone(f)\to X[1]$.

What is the closest analogue of this construction for a (more or less) general pointed homotopy category? My problem here is that we do not have to put any restrictions on $f$ in $C(A)$, whereas in model categories (co)fibration sequences are defined for (co)fibrations of (co)fibrant objects only. Certainly, there are model structures for categories of complexes for which all objects are (co)fibrant; yet being a (co)fibration is surely a restriction on $f$ even in this setting. Should one 'rotate' (co)fibration sequences?

Under which conditions one can prove that a filtered limit (or homotopy limit) of (co)fibration sequences is a (co)fibration sequence? Note that that the distinsuished triangles for cones commute with arbitrary(?) small limits (those that exist in $C(A)$; the existence of all such limits is determined by $A$).

For a spectral model category this limit question seems to be related with certain 'continuity of the enrichment'. Are there any terms or papers related to this property (or does it follow from some other properties or axioms for spectral categories)?