Well first of all the size of a black hole is defined by the event horizon. In fact the event horizon defines what a black hole is.
So we start from an event horizon, not from an infinitely dense point - that's not needed to form to a black hole.
We do need enough matter or energy (either or both) in a small region to form an event horizon. But that happens at a finite density. It does not have to be mass - just energy of any form.
The issue with "is there a singularity inside a black hole" is kind of tricky.
From outside the event horizon you have no way of knowing what's inside the event horizon. There's a concept called the no hair theorem which explains that you only get to know the mass, charge and rotational speed of the black hole as a complete unit. You can't know anything about the interior "structure".
So from the outside, the idea of "what is it like inside" has no useful meaning. From the outside the existence or not of a singularity makes no practical difference, in effect - the black hole will behave just the same with one as without.
Now inside the black hole, things are very, very different. All paths lead to the singularity in a classical black hole. Point anything at any speed in any direction and it will still end up at the singularity. And, from the point of view of someone inside the black hole, that will happen them in a finite time.
But inside and outside are effectively separate bits of the universe. They can't communicate and they don't experience the same things at all.
Now whether a singularity can really exist inside a black hole is a very debatable subject for theorists. To answer that question properly we would need a complete quantum theory of gravity - but we don't (at this time) have one. Best guess would be that the Heisenberg Uncertainty Principle would mean that at the core of the black hole we'd end up with a fuzzy region of uncertain make up and extremely high energy density, but not a real classical singularity, so (probably) no infinite density and no problem :-)