When I download the zip file, I can only open it with the 7zip. I extract to a location. Then select sqldeveloper.exe and a cmd prompt box opens for a second and disappears and nothing else happens. I hope this is more detail.
Try opening a Cmd window and running ...\sqldeveloper\sqldeveloper\bin\sqldeveloper.exe. That way any error messages will go to the console window and it will stay open for you to read it.
Usually downloading the 32-bit SQL Developer with the 32-bit JDK included works fine on both 32-bit and 64-bit Windows. Just make sure you have write access to the \Users\<your_userid>\AppData\Roaming directory: that is where the user-related SQL Developer settings are typically stored (unless somehow customized) on Windows 7.