Is one of these lenses better than the other in low-light conditions?
That's a close call. The 50/1.8 has a wider aperture, but only by 1/3 stop. That might translate into a tiny bit more shadow detail when shooting b&w films. However, I'd say that there's no significant difference.
And is it safe to say that the shorter a focal length the further things will appear, and a longer focal length means things will be closer?
Correct, although there's only a modest difference between these two focal lengths. A handy rule of thumb: shooting distance to get the same image size is proportional to the focal lengths. So comparing a 35 to a 50, for example, you could say that the 50 will show a given object at the same size from 5 feet that the 35 will show from 3.5 feet (because 5 and 3.5 are the same proportions as 50 and 35.) That's only a difference of a foot and a half, which you could achieve just by taking one step forward or backward. The difference gets more significant at longer distances -- 35 feet and 50 feet, for example -- but you can see that in the vast majority of cases, you could get the same image size with one lens that you could get with the other just by getting a little farther or closer.
Also, between short and long focal lengths, what is the difference in the depth of field? Does one have a more shallower DOF?
In general people will say that the wider lens will have greater depth of field, but that's because the wider lens provides less image magnification at a given shooting distance (the subject will look smaller.) In other words, focal length doesn't matter directly for DOF -- it's all about image magnification. If you take a picture of the same subject with both and enlarge the prints so that the subject is the same size in both, the DOF will be roughly equal.
[Warning: There are probably more arguments on photo forums about DOF than anything except bokeh and Canon vs. Nikon, so if you ask this question in six different forums, you'll probably get at least 60 different answers! The answer I gave you is technically correct, but since DOF is actually just an optical illusion, a lot of subjective factors come into play in different circumstances.]