For (3), here is a nice counterexample if we ignore contractibility (taken from an article from The American Mathematical Monthly):
http://www.jstor.org/stable/pdfplus/2695643.pdf
Hopefully someone can embed the image (Figure 13). It is two topologically distinct manifolds with the torus as a boundary... one being a solid [knot] torus, and the other being a 'cylinder' with a knotted inner-hole.