Safari is certainly the best browser to use to downld dmg's. If you hadn't been using it then I would have recommended it. Might I suggest that instead of double-clicking the dmg, you drag & drop the dmg onto Disk Copy. Open Disk Copy, go to its preferences (command & comma) & click the verifying tab, tick Ignore invalid checksums. Then drag & drop the limewire.dmg file onto disk copy icon in the Dock. It should open then with any luck. (Drag & drop is safer than double-clicking a dmg for some reason)

But upon a re-read of your post, it looks like you're having problems downlding it & you might be getting a corrupt downld or something. Try turning your firewall off if it's turned on (System preferences>Sharing>Firewalls & turn option off.)

Are you using dial up to downld it or a faster connection? Perhaps you should quit Safari, re-open & try again. Perhaps even a reboot. Other things to try, empty Safari's cache or/& reset it.