Firefox has nothing to do with a Big Fish game. But it sounds like you have to log into Zodiac online in order to start the game. If that's the case, the Capri game could be effected by your web browser.
The game exe for Big Fish games is a hidden file. If you tell Windows to make hidden files visible, the game exe will show up as the only hidden file in the game folder. Hidden files and folders will look sort of semi-transparent -- not as "solid" as regular files. If you don't see any semi-transparent files in the RealMyst folder, then you probably haven't displayed hidden files and folders and haven't set it to "Run as Administrator." Here's a site with links that tell how to make hidden files visible in different versions of Windows: http://pcsupport.about.com/od/tipstricks/f/show-hidden-files.htm
Using "Run As Administrator" on the desktop shortcut that Big Fish creates for you doesn't help because it points to the Big Fish Client and not the game executable itself.
You can configure the Big Fish client to install someplace other than C:\Program Files. Open the Big Fish Client, click "Options" at the top right, choose the Downloads tab, and at the bottom is a box where you can type in or browse to the location where you want the Big Fish Client to install your games. Then OK the change.
and about Capri...
Since I don't know exactly how Zodiac works, but Geo says there is no game exe in the game folder, I assume that they require you to run an executable that is on the Zodiac site instead of on your hard drive. If that's the case, there may be a conflict with your web browser, antivirus, or software firewall. Trying to run an executable from off the Internet may not be seen as "safe" by antivirus or software firewalls and may be being blocked. Zodiac may be trying to get around that by storing the game executable in a Temporary folder somewhere on your hard drive, but that may also be blocked. Your antivirus may even be deleting the file in the temporary folder as soon as it's created. And without knowing the name of the file, you can't create an exception in your antivirus.
I don't know what the .zdc file does or how Zodiac makes use of it, so I can't really answer your question.