You have a chunk of ideal rubber and can stretch and shape it any way you want, but you are not allowed to puncture it. The chunk of rubber will always be a ____.
shapes described by the term: ball, cup, wire, pyramid, comb, knotted shoelace, head of tangled and unbroken hair
shapes not described by the term: ring, cup with handle, chain, tee shirt
Sharp edges and points are allowed. Are there distinct terms for the surface and the solid?
Yes, there are different terms for the surface and the solid. The surface is called a sphere, and the solid is called a ball. A topological space homeomorphic to a sphere is called a topological sphere (Encyclopedia of Mathematics); a topological space homeomorphic to a ball is called a topological ball (Wikipedia). Topological equivalence is often described in terms of deforming rubber.
There are two different notions of equivalence of topological spaces: homeomorphic and homotopy equivalent. Solid spheres and all your examples in the first list are homotopy equivalent to a single point. Think of it as being able to "squeeze the shape all the way down." Spaces that are homotopy equivalent to a point are known as contractible.
Spaces that are not contractible include the examples you listed like rings and cups with handles as well as hollow spheres (By the way, usually the word sphere refers to hollow spheres, and solid spheres are referred to as discs or balls).