Do I have to smell the cork?
You're dining at a restaurant. You order a bottle of wine. The server or sommelier presents the bottle to you to make sure it's the right one, before he opens it. After he uncorks it , he hands the cork to you. What should you do?
A. Smell it--if it has an odd smell the wine is definitely corked, so send it back without even tasting it.
B. Smell it--if it has an odd smell it means the wine is definitely fine.
C. Give it a quick glance, and then set it on the table.
D. None of the above
All you really need to do is C--Give it a quick glance, and then set it on the table.
What can the cork tell you about the wine? Well, for one thing, if it's wet, then the wine has most likely been stored on its side (either that, or it's been jostled around on its way to the table). Storing a wine on its side is a good thing--it reduces the amount of oxidation taking place in the bottle, since less oxygen is coming in contact with the wine. But if the cork is bone dry, don't discount the wine, especially if it's young. Oxidation most likely hasn't taken place yet anyway.
Looking at the cork can also verify the wine's authenticity--the cork often has the name of the winery on it, so you know that what you ordered is what you are getting. (This used to be more relevant long ago when wine counterfeiting was commonplace. I'm not sure that you have to worry about your bottle of Screaming Eagle or Penfold's Grange being a fake...)
So what about smelling the cork? In my experience, a cork can smell funky and the wine in the bottle can be fine, or vice versa. So smelling it is not a good judge of quality. You'll need to actually smell--and taste--the wine, to judge that.
Feel free to take the cork home, though. A lot of wine lovers (myself included) collect them, and fill vases, bowls, baskets, etc. with the corks from bottles they have enjoyed.