|MadSci Network: Astronomy|
To start, it depends on what you mean by "stop spinning." If you mean not spinning with respect to the stars (which are basically fixed in our sky), the answer is yes, it is possible a really well placed collision could stop us. In fact, if you look at the physical properties of the planet Venus, you'll notice it takes 243 days to rotate once. It is believed that this incredibly slow rotation could be due to a collision early in Venus's history.
However, odds of completely cancelling out our rotation are very low. It would take a large body, nearly planet sized at least, to cancel out the rotation of Earth or Venus, and there aren't any asteroids big enough for that anymore. In the case of Venus, it was probably hit early in the life of our solar system, when there were many more large bodies orbiting our sun.
If you wonder instead if we might ever slow our rotation enough so that we always show the same face at, say, the Moon just as the Moon shows us the same face, the odds are much better. Due to tidal forces (the same forces that deform Earth's oceans and raise tides), when a body does not always show the same face to the body which creates its tides, it feels a braking force, slowing it down. This is what has happened to the Moon, and in fact the Earth has had its rotational period increase from 14 hours to 24 hours thanks to the Moon.
Getting the Earth to always face the sun would be much more difficult as long as the Moon is still here. This is because the Moon is the main driver of tides on the Earth. If you could get rid of the Moon and wait long enough, Earth would eventually always face the sun. [Conversely, if you could get rid of the Sun, the lunar tides would eventually cause the Earth always to show the same face to the Moon. Moderator]
Could we ever get Earth started again? I don't want to guess what kind of technology we'll have in millions (or billions) of years, should Humanity still be around, but presently there is nothing we could do to spin the Earth up. It's just way too big for that. A larger asteroid (again, nearly planet sized) colliding with the Earth might be able to give us spin again, depending on how it hits, but it would alway wreck Earth's ability to support life for a very long time afterward.
As far as the environment of Earth were it always facing the Sun: it may not be as bad as you think. As long as air still circulates, the dark side could stay unfrozen, or at least not too cold. Venus, even with its slow rotation, has strong winds to circulate air around the planet, keeping the temperature fairly constant. I'm not aware of any models for the case of a non-rotating Earth (climate models are notoriously complex and take a long time to run), but it may not be unreasonable to expect a somewhat similar situation.
For more information on planets and tides, I recommend Moons and Planets by William K. Hartmann. This is a college-level text written by one of the great experts on planets. If I've piqued your interest in Earth's twin, Venus, I recommend Venus Revealed by David Grinspoon. While Grinspoon is an expert on Venus, this is not a textbook and should be a good read.
Hope this helps!
Try the links in the MadSci Library for more information on Astronomy.