Concerning your last question, I would say you should view your C-algebra itself as the non-commutative topological space. The spaces you suggest are commutative. Your C-algebra might be considered as a non-commutative analog of one of them.

Concerning your last question, I would say you should view your C-algebra itself as the (coordinate ring on the) "non-commutative topological space." The spaces you suggest are commutative. Your C-algebra might be considered as a non-commutative substitute of the coordinate ring on one of them.