Hi John,
I've not had a chance to investigate further as my day job (teacher) has been hectic as we are trying to finish courses and get ready for assessments!
Any features for pdf files are stored in the non OpenSongApp database file you found, which should be persistent between uninstalls/reinstalls. However, due to Android storage issues (the bain of my life), this database isn't always updated (which is why it might show with 0kb) The actual database is in the Android/data/com.garethevans.church.opensongtablet/ folder (which is cleared on uninstall). The database matches the song file name and folder location of the actual pdf, so when that is loaded, it looks in that database for a matching record. If the file name is changed, that will break that link.
Apostrophes are an issue in sqlite databases and I thought I'd encoded them to avoid this issue, but it sounds like something is going wrong with that. It won't be anything you are doing.
To be honest, the current version of the app isn't great for non OpenSong formatted songs and this database was an after thought with many hack workarounds to implement it. I've actually been trying the app to make it much slicker, material designed and loads more features, but progress is slow. The non OpenSong stuff works way better in this too. Unfortunately it is still pre alpha though!
I will try to get a look at this issue for you, but reinstalling, etc won't fix it, it will be a coding issue at my end.
To export sets, etc. You can manually do this for individual sets using the app, but to back up all of them, this isn't currently a feature. You can, however, copy the contents of the sets folder though.
I tend to sync my devices using Dropbox/Dropsync as described in the user guide. This syncs songs and sets (but won't sync non OpenSong databases of they show with 0kb).
I'm trying not to spend too long fixing less well used features in the current version of the app just now as this restricts my progress in the pre alpha version (which does have all of these fixes in place).