Let $c^i: M^{\ge i+1}\to M^{\ge i}$ for $i\in \mathbb{Z}$ be an sequence of morphisms in a triangulated category $C$ and assume that $M^{\ge i}$ are equipped with compatible morphisms into an object $M$ of $C$. I would like to relate the (co)homology of $M$ with that of $M^{\ge i}$ and also of $M^i=Cone(c^i)[i]$. So, for $H:C\to A$ being a homological functor ($A$ is an abelian category) I take an exact couple with $D_1^{pq}=H(M^{\ge p}[p+q])$, $E_1^{pq}=H(M^p[q])$, and connecting morphisms coming from (triangles corresponding to) $c^i$.I have two questions related to this couple and the corresponding spectral sequence.
What is the best reference for the following fact: if $M^{\ge i}=0$ for $i$ large enough and $=M$ for $i$ small enough then this spectral sequence converges to $H(M[p+q])$? I only know the exercises after §IV.2 of Gelfand and Manin's "Methods of homological algebra" (where Postnikov systems are treated; the notation there is different and possibly there are some indexing mistakes).
If there are no extra restrictions on $M^{\ge i}$ then the spectral in question does not necessarily converge to anything. However, the morphisms $M^{\ge i}\to M$ certainly give certain filtration on $H_{\ast}(M)$; more generally, for any $j\ge 1$ and $D_j^{pq}$ being the corresponding term of the $j-1$th derived couple there is a canonical morphism $D_j^{pq}\to H(M[p+q])$. Does there exist any formalism for treating this situation (some "very weak convergence" of spectral sequences notion)?
An example: if $C$ is the homotopy category of complexes $K(B)$ ($B$ is an additive category) then one can take $M^{\ge i}$ to be the corresponding stupid truncation of a complex $M$ (so, we put zeroes in degrees less than $i$); then $M^i$ is the $i$th term of $M$. If $M$ is a bounded complex then the additional conditions of my first question are fulfilled.
