EDIT After thinking carefully with the help of the clear answer of ZhenLin, I think I will reformulate my question the following way. The text of my original question is kept below.
The claim of Prof. Rotman in section 10.5 that split complexes (not necessarily exact) with projective terms are the projectives in the category of chain complex relies on the result of exercise 10.15.
In this exercise, it is asked to prove that if $$0\to A'\to A\to A''\to 0$$ is split exact with $\delta$ the injection $A'\to A$, then the complex $$0\to A'\to A\to 0$$ concentrated in degrees $k$ and $k-1$ with the only non zero differential $\delta$ is a direct summand of the complex $$0\to A\to A\to 0$$ concentrated in degrees $k$ and $k-1$ with the only non zero differential $1_A$.
This exercise looks innocuous but I cannot solve it and I believe it is wrong.
If I am not mistaken, this explain why the approach of Prof. Rotman is wrong, that is to say the projectives in the chain complex category must be split exact complexes, not only split ones. Therefore Corollary 10.37 & Theorem 10.42 are wrong, and also the fact that Cartan-Eisenberg resolutions are projectives ones!
But perhaps I am confused. Does somebody can give me his informed opinion here?
In Prof. Weibel AIHA book exercise 2.2.1 it is asked to show that the projectives of the chain complex category of a category of modules are the split exact complexes of projectives.
In Prof. Rotman AIHA book theorem 10.42 it is proved that the projectives of the chain complex category of an abelian category are the split complexes of projective, not necessarily exact! There are some points in his proof that I do not understand:
1/ He uses the Freyd-Mitchell embedding on an infinite numerable direct sum (Prop. 10.36). Is it legitimate ? Edit: there should be a workaround here because locally in each degree of the complex the infinite direct sum is finite.
2/ In his proof of Corollary 10.37, he said that direct sum of projectives are projectives. Is it true in any abelian category ? Edit2: yes, clearly as Zhen Lin pointed out in his answer.
If we work in a module category, I can understand his proof. So who is right ? Edit3: there must be something wrong in the proof even in a category of modules, since the answer of ZhenLin is crystal clear. I have to look in details at it.