Can the Finder resume a file copy? I know about rsync and how to use it, but sometimes I just want to copy files using the Finder. However, if that fails, I have to do everything all over again, since the Finder does not seem to be able to resume a file copy action. Is there a way around this?
No, the Finder does what it does and that is do its best not to leave anything unfinished.  If it can't finish a copy it deletes the partials.  

You could probably hack together a scriptlet that would let you drag and drop files onto it that would then rsync the file somewhere. You could even make it a folder action.  But it would be limited in where the files end up.