As it came out in the comments, your doubt about cellular cohomology of $\mathbb{RP}^n$ not being isomorphic to singular cohomology was because you switched the even and odd cases in $\partial$, as seen on Hatcher (current online edition) p.144.
In fact, cellular (co)homology is always isomorphic to singular (co)homology for CW complexes: Hatcher is again a good reference for this, see pages 139 for homology and 203 for cohomology. The proof for homology doesn't involve chain maps, quasi isomorphisms or chain homotopies, rather it hinges on an ad hoc diagram chasing argument, in which one uses the fact that the chain groups of the cellular chain complex are the relative singular homology gruops $H_i(X^i,X^{i-1})$. With $\mathbb{Z}$ coefficients it is true that there is a quasi isomorphism (indeed a chain homotopy) of the two complexes, since it os a general fact that for complexes of abelian groups, isomorphic homology is a sufficient condition for there to exist a chain homotopy (see here). I'm not sure whether this holds for some larger class of modules (like modules over PIDs), but for chain complexes of modules over a generic ring this is false in general, and I'm not sure if it turns out to be true in the case of singular and cellular complexes of CW complex (this is what the OP seems to be asking here).
Again, the standard proof of the equivalence of singular or cellular cohomology doesn't operate with maps at the level of chain complexes, and is similar to the proof for homology, to the extent that the bulk of the argument is just handled by using the universal coefficient theorem and the homological case, if I remember correctly. Hatcher not only shows that $H^\bullet(X;G) \simeq H^\bullet_{CW}(X;G)$, but also that the cellular cochain complex is the dual complex of the cellular chain complex, which isn't evident from the definition.
I hope all of this answers your questions. I could add details, but Hatcher is really a very good reference for all of this.
In the case of a field, this is true. We can write $Z = B\oplus i(H)$ for $i : H\to Z$ a section of the projection and $C = L\oplus B\oplus i(H)$. Define $j : H\to C$ using $i$, and define $q : C\to H$ by sending $L$ and $B$ to $0$, and projecting $i(H)$ back.
These are maps of complexes: $qd = 0 = dq$ since $H(C)$ has trivial differential, and $jd=dj=0$ since $j$ lands on cycles. Moreover, by construction, $qi = 1$.
Finally define $h : C\to C$ by picking a section $s : B\to C$ of $d : C\to B$ and sending
$b\in B$ to $s(b)$, and $L$ and $i(H)$ to $0$. Then,
- If $l\in L$, then $(hd+dh)l = l = l-iql = (1-iq)(l)$,
- If $b\in B$, then $(hd+dh)b = dsb = b = (1-iq)(b)$,
- If $z\in i(H)$, then $(hd+dh)z = 0 = (1-iq)(z)$,
so $dh+hd = 1 - qi$ is a homotopy between $1$ and $iq$.
Best Answer
You will find this kind of result in
Blakers, A. "Some relations between homology and homotopy groups". Ann. of Math. (2) 49 (1948) 428--461.
I am pretty sure it is in Massey's book on Singular Homology, from a cubical viewpoint.
Proposition 14.7.1 of Nonabelian Algebraic Topology gives a deformation of the singular cubical complex of a space onto that coming from a filtration, under conditions which are satisfied in the case of a cellular filtration.
Later: Here is the detail of the proposition. For the question you can assume $X_*$ is the skeletal filtration of a CW-complex and $R X_*$ is the cubical set of cellular maps $I^n_* \to X_*$:
Let $X_*$ be a filtered space such that the following conditions $\psi (X_*, m)$ hold for all $m \geqslant 0$:
$\psi (X_*, 0) :$ The map $\pi_0 X_0 \rightarrow \pi_0 X$ induced by inclusion is surjective;
$\psi (X_*, 1) :$ Any path in $X$ joining points of $X_0$ is deformable in $X$ rel end points to a path in $X_1$;
$\psi (X_*, m) (m \geqslant 2 ) :$ For all $\nu \in X_0$ , the map $$\pi_m (X_m , X_{m-1} , \nu ) \rightarrow \pi_m (X, X_{m-1} , \nu )$$ induced by inclusion is surjective.
Then the inclusion $i \colon RX_* \rightarrow KX=S^\square X$ is a homotopy equivalence of cubical sets.
The proof is quite direct by induction because the relative homotopy groups may be defined by maps of cubes, and in cubical sets, homotopies are defined using cubes.