Yes. But here's a weird thing. When I load Skype (I'm still not using it, trying to learn how to use it...) I see that it mounts the disk image on my desktop. It loads ok and works ok AFAICS, but it does that.
I've seen that quite frequently on machines checked in for service. Skype and Firefox are the big two, but I've seen it with other apps also.
Mac is "weird" (unusual) in that it can generally run apps from locked volumes, and does not require running an installation process in most cases. So you can download FF and run it exclusively from the disk image, never actually having installed it. And many people do.
But it's better to install it. That way it can run auto updates, you're not bothered with a disk image on your desktop, and don't happen to delete the mother DMG file wherever it happens to be (Downloads probably) because you don't realize you still need it and are using it.
