I used to use Zend Studio (which is now built on Eclipse); but recently I switched to Linux at work, and haven't installed ZS yet. I've used nothing but the ViM editor.
The "run" button is for debugging, it's not for previewing the webpage. You would have to install Zend's debugging extensions into your local apache config. Of course the alternative is to just go download Zend Server and install that (it's preconfigured for debugging I believe).
I never got debugging working right with Zend Studio when I was working in Windows. So I just set up a separate development server, and set display_errors to on, and set the error level to it's highest (E_ALL | E_STRICT). No need for a debugger then.
Sorry I couldn't be much help. The only thing I could tell you is to look into Zend Server (or Zend Platform, or both). They cost money; however, you can still use the trials for debugging once the trial runs out. Or at least you could with the last version.