In the normal rules of euclidean geometry, it would seem you would find a center of a finite universe. But remember that we aren't dealing with Euclidean geometry - we are dealing with the curved spacetime of general relativity. Without learning Non Euclidean geometry, you can use the analogy of any curved 2D surface to see how the universe could be without a center. Often, surfaces such as that of tables are flat, but they can also be curved as in the case of a balloon. In that case, at no X,Y point will you be able to find a center. The analogy has its shortcomings, mainly because the universe is not the curved area of a surface, it's a curved volume.
Furthermore, the term "fabric of spacetime" is somewhat of a popular physics word that isn't meant to be taken too literally. Spacetime isn't considered to be an aether like medium, and there is no actual substance that empty space is consisted of. Following from that, the universe should be in no danger of tearing from the expansion.