The analogy about placing a mass on top of a sheet of paper is pretty problematic, for precisely this reason. No, gravity does not create extra dimensions, at least not in any scientifically meaningful way. Another common question that this analogy leads to is the following: if spacetime is expanding, what is it expanding into?
It is crucial to understand that curvature of a manifold is an intrinsic property of that geometry, that does not need us to embed the manifold in a higher dimensional Euclidean space. It is studied by measuring distances between points on the manifold itself. A better analogy would be the following: take a sheet of paper, draw two points on it ($P=(x_1,y_1), Q=(x_2,y_2)$). Now, instead of considering the distance between them to be $$PQ=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$ define some other notion of separation between points (define a different metric). This would allow us to define a manifold that is topologically the same as a sheet of paper, but has different local geometrical properties. What is important is that at no point do we need to care whether the paper is floating around in a three dimensional room or a fifteen-dimensional one. All questions about the geomtery can be answered within the two dimensions of the sheet.
The above might seem like a matter of interpretation, which would mean that the analogy is fine. However, this is more than just a matter of semantics. If we assumed that we were fixed on a three-dimensional spatial slice that is a subspace of a four-dimensional space (and this has been done in braneworld scenarios), gravity sees all the dimensions, including the larger space, and the physical effects in our world would be vastly different (for example, the gravitational interaction would fall off much faster). If on the other hand, we assumed that nothing sees the extra dimensions, (i.e., there is no way to tell if we are embedded in a five-dimensional spacetime), then it becomes scientifically meaningless to postulate their existence.
