1) Building the UN makes you eligible to be elected Secretary-General, which lets you propose resolutions. However, all Civs get a vote based on total population in their Civ. If you don't want to participate in the resolutions, vote Abstain, but if you do so, you are forced to follow the decision of the UN. If you hate it, but don't want to vote no, Defy the Resolution to make it fail automatically.
2) Cam_H explained it well.
3) Nukes. Ahh, the weirdest (to me) part of Civ. Basically, ICBMS and Tactical Nukes (BTS only) can be used to Nuke a city. First, the game checks to see if the Nuke will be intercepted; i.e, if you have built the SDI Project, there's a 75% chance that any nuke aiming for you will be intercepted and destroyed. If the Nuke is not intercepted, it hits the city, creating the wonderful Mushroom Cloud. It also converts the 8 squares around the city to "Fallout" terrain, which can be removed if you have research Ecology. Next, the game reduces the city's population by a certain amount (not sure how that works). Finally, the game performs a "killcheck" on each building in the city. World Wonders and National Wonders survive automatically, but every other building has a probability of being destroyed. If the building is destroyed, of course, it won't be in the city and you'll have to rebuild it. Also, using a nuke gives diplomatic penalties.
Each Nuke used will result in a -1 with the Nuked Civ, -1 with Civs Pleased or Friendly with that Civ, and may result in nations declaring war on you..