Does a black hole necessarily contain a singularity?
Yes, an event horizon will form once escape velocity exceeds the speed of light. It does not necessarily require a singularity inside it.

However, there is no known physics that would allow any non-point-like object to exist inside an event horizon. It is hard to see how any traditional forces would be able to support an object in such case, and for quantum mechanics we simply do not have the theory that'd be valid in such high gravitational field.