1. Mikhail, mapping cones etc, are defined for arbitrary maps. The problem is that they are not homotopy invariant unless your model category is left proper. Therefore, in general you must take cofibrant replacements etc. Complexes form a left proper model category with the projective and with the injective model structures.

2. Homotopy colimits of cofiber sequences are cofiber sequences. If you want arbitrary filtered colimits of cofiber sequences to be cofiber sequences, the most usual hypotheses are set theoretical: your model category must be combinatorial. Then there is a cardinal $\alpha$ such that $\alpha$-filtered colimits of cofiber sequences are cofiber sequences. The cardinal $\alpha$ is the smallest one such that your category is locally $\alpha$-presentable and has sets of generating (trivial) cofibrations with $\alpha$-presentable sources (and maybe targets too, I don't remember). E.g. for simplicial sets you can take $\alpha=\aleph_0$.

I don't know of how bein spectral may simplify my answers above. Referecens for 1): Hirschhorn's book rather than Hovey's. For 2): you have to look at papers on combinatorial model structures (Dugger, Rosický...), as books don't cover well this part of the theory.

1. Mikhail, mapping cones etc, are defined for arbitrary maps. The problem is that they are not homotopy invariant unless your model category is left proper. Therefore, in general you must take cofibrant replacements etc. Complexes form a left proper model category with the projective and with the injective model structures.

2. Homotopy colimits of cofiber sequences are cofiber sequences. If you want arbitrary filtered colimits of cofiber sequences to be cofiber sequences, the most usual hypotheses are set theoretical: your model category must be combinatorial. Then there is a cardinal $\alpha$ such that $\alpha$-filtered colimits of cofiber sequences are cofiber sequences. The cardinal $\alpha$ is the smallest one such that your category is locally $\alpha$-presentable and has sets of generating (trivial) cofibrations with $\alpha$-presentable sources (and maybe targets too, I don't remember). E.g. for simplicial sets you can take $\alpha=\aleph_0$.

I don't know of how being spectral may simplify my answers above. Referecens for 1): Hirschhorn's book rather than Hovey's. For 2): you have to look at papers on combinatorial model structures (Beke, Dugger, Rosický...), as books don't cover well this part of the theory.