The eyepiece cover isn't really used often but it does serve an important role. When metering very dark scenes or when using a strong ND filter, light entering the viewfinder can interfere with the meter reading because the metered light "through the lens" is so dim.
So you use the eyepiece cover to prevent that from happening, and get an accurate exposure.
It is also possible that light entering the viewfinder can actually get into the lightbox while an exposure is being taken. For that reason, people often leave the eyepiece cover on when actually taking the picture too (again, when using long exposures in bright daylight such as when you use a ND filter).
In my experience, I have not had problems with this except when using my 10-stop ND filter. Then I use the cover, or more accurately, I cover the viewfinder with my thumb or some other object. some people have fashioned little "drapes" that attach to the camera hotshoe and then drape a small piece of cloth over the viewfinder, in lieu of using the official cover. This is because on most cameras you have to remove the rubber eyecup to install the cover, which is fiddly and tedious. Higher Nikons (Canons too?) have a viewfinder shutter as snberk says, by flipping a lever.
Re: neck straps- I don't really use them either. When walking with a backpack I use the Op/tech backpack straps that apply the weight of the camera to the backpack shoulder straps instead of around my neck. When shooting I typically just hold my camera in my hand. Actually I find the op/tech strap system to be very versatile, because it has in-line connectors that allow you to add various types of strap to your camera with ease. For example I have a neoprene neck strap that I can also swap for the above mentioned backpack straps, and also two short lengths of strap that I can leave on and use as a wrist strap if I want as well (these are actually the adjustable portion of my neoprene neck strap). And when I want no strap, I just disconnect everything and am left with two small clips on my camera body, which keeps clutter down.
I don't leave straps on my camera when using a tripod as I feel it just gets in the way and is a potential to get snagged/caught on something and tip the tripod over, or pull it out of my hands, etc.