If I delete a paid app purchased through the Mac App Store, can I redownload without paying?

Solution 1:

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.