Turning Lee's comment into an answer:

Compactification theorem for differentiable manifolds ?

In the first paragraph you are asking if all manifolds a tame. The answer is "no" and there are even simply connected examples, due to Whitehead. The second half of your question is very different, and seems to be asking about smoothing corners.

Turning Lee's comment into an answer:

In the first paragraph you are asking if all manifolds are "tame" (http://en.wikipedia.org/wiki/Tame_manifold). The answer is "no" and there are even simply connected examples, due to Whitehead. The second half of your question is very different, and seems to be asking about smoothing corners.