If a polyhedron is homeomorphic to a simplex, is it piecewise-linear homeomorphic?

If you assume that your polyhedron has only finite number of faces, I think the answer to your question is unknown. Moreover any answer to such question would give a solution to Smooth Poincare conjecture in dimension 4, which is still open.

Indeed, suppose you have a four-dimensional sphere with an exotic smooth structure. Then you can always triangulate such a sphere in a finite number of simplexes. Now, throw away a simplex from such a triangulation. What you get is a homeomorphic to a simplex, but can not be PL diffeomorphic to it, otherwise your initial sphere would be PL diffeomerphic to the standard one, which is not sow since you sphere is exotic.

It is not true in dimension 5 or above. Edwards' double suspension theorem implies this. See this Wikipedia article section.

I am not sure if this is what you are asking, but check out http://en.wikipedia.org/wiki/Exotic_R4 (note that in dimension four, PL is the same as smooth).