Is it possible to search for existing songs across multiple projects, without looking in each project manually until found?
Yes, when you're adding songs to a set list, or quick-adding a song while viewing a set list, you can click the search button in the top toolbar and turn on the Include All Projects filter.
It would nice for this to work in all occasions not just set lists. There are times when songs seem to get removed from a project and I can't find them.
I can add that to my wish list.