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(Co)chain complexes, abelian Categories, (pre)sheaves, (co)homology in various (possibly highly generalized) settings, spectra, derived functors, resolutions, spectral sequences, homotopy categories. Chain complexes in an abelian category form the heart of homological algebra.

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Is the functor $mod(Tw\mathcal{C})\rightarrow mod(\mathcal{C})$ cohomologically full and fai...

This seems to me to be unproblematic (if messy to prove). Try if this does the job: Proposition 12.15(2) in Bespalov-Lyubashenko-Manzyuk, Pretriangulated A_infty-categories
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