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

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?

2. 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 $A$).