Are new galaxies able to form on the edge of the expanding universe?
It seems that you are thinking of the universe as expanding out from a central point in three dimensions and therefore having some kind of edge.
As far as we can tell this is not the case.
A lower dimensional analogy might be the surface of a balloon that is being inflated. There is no central point on the surface of the balloon, nor is there an 'edge' to the surface - at least from the perspective of the surface.
So our own universe could be pictured as a three dimensional hypersurface expanding in a four dimensional space. Note that this is just a picture - there is no four dimensional space into which our universe is expanding (as far as we know). However, it does help understand how our universe can be expanding, but not have a centre or edge.
To attempt to answer your question another way, we might ask whether new galaxies are still forming. The answer seems to be yes. Not only dwarf galaxies, but major galaxies still seem to be forming - according to NASA's Galaxy Evolution Explorer, which has observed much younger galaxies than our own Milky Way.