Topological $n$-manifolds have the homotopy type of $n$-dimensional CW-complexes I search for a chain of clean references, which lead the fact of topological manifolds of dimension $n$ having the homotopy type of a CW of dimension $n$.
Milnor's On spaces having the homotopy type of a CW-Complex proves that every topological manifold has the homotopy type of a countable CW-complex since it is an absolute neighborhood retract.
Theorem E of Wall's Finiteness Conditions for CW-complexes then gives me the desired answer for dimensions at least 3, as long as I know that the universal covering $\tilde{M}$ of a topological manifold $M$ of dimension $n$ has vanishing homology up to dimension $n$ and it holds $H^{n+1}(M,\mathcal{B})=0$ for all abelian coefficient bundles $\mathcal{B}$.
Theorem 3.26 and Proposition 3.29 of Hatcher's book gives me the first claim about the homology since universal coverings of topological $n$ manifolds are again topological $n$ manifolds since the fundamental group of a topological manifold is countable (a clean reference for this is Theorem 7.21 in Lee's book 'Introduction to Topological Manifolds')
Since I am happy to forget about the low dimensional cases, it leaves me with the search for a solid reference to the claim

Let $M$ be a topological manifold and $\mathcal{B}$ be an abelian coefficent bundle, then $H^{n+1}(M,\mathcal{B})=0$.

 A: Vanishing of cohomology above the top dimension follows from the Poincare duality, which is proved in a general form in Bredon's "Sheaf theory", Chapter 5, section 9. This works for all topological manifolds (compact or not) and for the local coefficent bundles as in Wall's paper (recast in sheaf theoretic language). 
A: Every topological manifold has a handlebody structure except in dimension 4, where a 4-manifold has a handlebody structure if and only if it is smoothable.  This is a theorem on page 136 of Freedman and Quinn's book "Topology of 4-Manifolds", with a reference given to the Kirby-Siebenmann book for the higher-dimensional case.  It is then an elementary fact that an $n$-manifold with a handlebody structure is homotopy equivalent to a CW complex with one $k$-cell for each $k$-handle, so in particular there are no cells of dimension greater than $n$. At least in the compact case a manifold with a handlebody structure is in fact homeomorphic to a CW complex with $k$-cells corresponding to $k$-handles; see page 107 of Kirby-Siebenmann.  This probably holds in the noncompact case as well, though I don't know a reference.
