Brief summary of simplicial, CW and manifold notions

Question

I tried to summarize the relations between the following notions of: a manifold (smooth, topological and PL), simpilicial complex, CW complex. However I found some inconsistencies, which may be not a problem for a specialist who knows how to interpret certain assertions, but I'm not a specialist so I'm trying to pose statements precise. So I would like to pose some statements, asking whether they are true (surely some of them follow directly from the definition but there are many conventions and I'm a bit confused about them):
1. The manifold homeomorphic to simplicial complex is usual referred as `admiting triangulation' 2. a) Every PL manifold admits triangulation
b) every triangulated manifold is CW-complex
c) there are CW complexes which cannot be triangulated
3. Smooth manifold is PL so in particular can be triangulated and is CW complex
4. a) Any compact topological manifold of dimension $\leq 3$ can be triangulated
b) for any $n >3$ there are compact topological manifolds of dimension $n$ which can not be triangulated (known earlier for $n=4$, quite recent for $n>4$
5. a) Any compact manifold of dimension $ \neq 4$ is homeomorphic to CW complex
b) for $n=4$ this is an open problem