I believe they would overwhelm a limited area downwind of the site(s) with radiation, but would only bring trace increases to the vast majority of the world. There would be localized problems in the affected area(s) for years, with contamination of crops and the water supply, but the idea of the whole planed being irreparably poisoned by a limited nuclear strike is simply not true. Too bad for the folks in the immediate vicinity, though.
Depends. If Cobalt-60 Bombs are used and in larger quanities, they can kill the planet. Their half-life is long enough that enough particles with enough radiation can encircle the globe. As far as I know, the US doesn't have any of these. Russia might however. Our standard fusion bomb in the high hundred+ Kiloton to megaton range is pretty clean(as far as nukes go anyway).
Cobalt shelled fusion bombs are called "salted"...the more Cobalt you "salt" on to the shell the longer that part of the Earth is going to be unihabitable(salted) after you light this thing. Cobalt-60 emits Gamma for 5 years minimum. Very very dirty. None have ever been tested as far as I've ever heard.