Yes. This is contained in Section 1.3 of this paper by Barthel, May, and Riehl. Proposition 1.14 proves that h-cofibrations are the same as degreewise split monomorphisms and h-fibrations are the same as degreewise split epimorphisms. The introduction defines the notation $M_R$ for the category of unbounded chain complexes of $R$-modules. Remark 1.16 discusses the idea that there should be two model structures, but they happen to coincide. This paper says the model structure is originally due to Christensen and Hovey, Cole, and Schwanzl and Vogt, all around the same time.

Yes. This is contained in Section 6 of this paper by Christensen and Hovey. In the bounded case, any projective class gives rise to a model structure. In Section 1.4 they introduce a projective class whose weak equivalences are the chain homotopy equivalences. They don't discuss having two such model structures, but using this you could check your "wild guess" by hand.