If I put a Nikkor 35mm f/2 FX lens on a DX camera, is it still 35mm f/2, or has the effective focal length or minimum aperture changed, because it is being used on a smaller format than what it was designed for?
Aperture is unaffected.
The field of view becomes 35mm x (the crop factor of your sensor), which is 1.5 in the case of Nikon DX cameras (It's also 1.5 for Pentax and Sony, 1.6 for Canon and 2 for Panasonic & Olympus).
So your 35mm Nikon lens has a 35mm x 1.5 = 52.5mm "35mm equivalent" field of view.
Note that the perspective doesn't change, just the amount of 'stuff' you can fit in the frame. The image will still look like a 35mm image as shot on a full-frame sensor, but with the sides cropped in to give you the field of view of a 52.5mm lens on a full-frame sensor.
(Most people round that to 50mm "35mm equivalent" field of view, regardless of 1.5 or 1.6 crop factor)
The post by Scott is a nice summary, but it contains a common misunderstanding - that the perspective doesn't change, which it does. The angle of view of the lens is multiplied with ca 1.5 (or 1.6 if you use Canon, or 1.3 if you use an expensive Canon), which also changes the perspective as much as if you'd used a 52.2 mm lens.
I used a Nikkor 35 mm f2 extensivly as a 'normal' lens, and it was really nice. Then I bought a Nikon 35 mm f1.8 and it was even better.