Does the iPod Classic respond to the clicker buttons on the iPhone earbuds?
Yep. Check out this link on Apple's site. I'm actually surprised to learn so many devices support it now. 


iPhone 4  
iPhone 3GS  
iPod touch (2nd generation)  
iPod classic (120GB 160GB - 2009)  
iPod nano (5th generation - video camera)  
iPod nano (4th generation - video)  
iPod shuffle (3rd generation)  
iPad