It depends upon the lens (if it's a prime), and/or the focal length (if it's a zoom). I frequently use the excellent 35mm f/1.8 DX lens in FX mode on my D700. There is certainly some fall off at the corners, but it works well for my uses (location portraits, and indoor people events).
To answer the question, YES, DX lenses can be used in FX mode. You can turn off the auto DX crop and assign it to the func button if you want to. You can crop the picture later if you want. Even though the corners of some lenses will be shaded, you can get a neat looking square crop. You can have a lot of fun experimenting. On my Tokina 12-24mm F4, I can use it as wide as 17mm without vignetting. However, on my Tamron 17-50mm F2.8, it vignettes at all focal lengths.
yes and no. It will "work" but the camera will be shooting in DX mode, so a D600 will be about 10 MP and D800 about 16 (similar in photo quality to d7000 (tests done) ) If you have a 12 mp FX camera it is only about 3MP. So if you need it to work, and you want that 1.5X crop factor then yes. if you want the 1X factor and full megapixels then no.
Yes you can use a DX lens in FX mode, but depending on the lens, you will have moderate to severe vignetting. There woould be no point use the 35mm DX in FX mode because you would have to crop your picture to the size of a DX shot, but the excellent Tokina 11-16mm lens works quite well at 16mm in FX mode.
depends on the lens. Most DX lenses will vignette (black ring around circumference of photo). But, vignetting aside, there will typically be severe distortion near that vignetting as well, and great drop in sharpness outside of the DX crop area. I've used the 35 mm DX 1.8 G on the D600 and results are so-so, it is okay for artistic effect, but gets old quickly and cropping it to DX is pointless since you can just get the 50 mm 1.8D for half the price and FOV will be equivalent, and there will be no vignetting.