For my Canon my primary lens is a 24-105mm f/4L because it's a very nice, light, general use lens. I have a 50mm f/1.8 prime though and I like it, especially for the price. The next lenses I'm looking to get are a 70-200mm f/4L IS USM and a 300mm f/4L IS USM to retire my current telephoto, a 55-250mm f/4-5.6 IS.
The Nikon 35mm F1.8 AF-s is a nice inexpensive walk around lens for normal perspective on a crop sensor body. However, if depth of field effects or portraits are more to your liking you would get more out of the 50.
I greatly enjoy the 50mm I have for my D3100, though my lens is only MF (way cheaper than the AF version that was available at the time). I usually carry my standard 18-55 and 50mm with me, though I've come into the habit of bringing my 55-200mm with me more places (but it feels really big). I would certainly advise getting it; the shots I can get with it are quite different (at times) from when I use other lenses, in a noticeable way.
Yes and no. Yes becaue with the AF-S, I can get a certain amount of clarity that I can't otherwise, but by not having AF-S, it creates a challenge for me to get just the right shot that I want. So I, ultimately, consider it a sort of virtue and a vice.
In terms of 50mm versus the 35mm, I would recommend the 50mm. True, the 35mm will give you roughly the same framing as a 50mm on a full-frame, but the 35mm still shows some of the qualities of a wider lens. The Nikon AF-S 35mm f/1.8 is the only option I can think of that you might consider for your camera, and its 40mm and 50mm cousins generally perform better.