To be clear, when they say faster than expected, they mean that the direct measurement of the recession gives a faster rate than the more indirect (but highly accurate) rate determined by doing a fit of a cosmological model to the measurements of the details of the cosmic microwave background that is remnant from the Big Bang. This has been known for a while, but now there is less chance that it is a fluke. There's of course already dozens of papers proposing new particles and interactions that can solve this discrepancy, but it will be a while before any of these models can be experimentally tested.
Either space is infinite or if you keep going in one direction you eventually go back to where you started, like the surface of the Earth (observations favor the first option). An actual "edge" is very unlikely.
Initially people thought it was just the "initial momentum" as you say, in which case, since gravity is attractive, the expansion would decelerate and could eventually reverse. At the beginning of this century, it was found that not only it is not slowing down, but actually accelerating, which is only possible if there is an energy density in empty space that repels gravitationally, instead of attracting, which was called the dark energy. It makes up about 70% of the total energy of the Universe right now, and nobody really knows what it is made of. If this dark energy is unchanging in time, the universe will expand forever. In some models, which don't look very likely right now, however, it can slowly weaken, in which case, the expansion could one day slow down and reverse.
