If I delete an app purchased through the Mac App Store, can I download it again without paying for it again like I can with iOS? In the event of an app restore, does the app retain settings, saved progress, etc from the previous installation?
Yes. Just go to the "Purchases" tab and click "Install" by the app you wish to reinstall:
Well-behaved apps should retain settings, etc upon reinstall, as merely deleting the app from /Applications is not sufficient to remove that information from your system. This may not be the case if an app cleaner is used.