Reading Settings -> Front Page displays “Static Page” being reset

I would suggest first to update your WordPress, so that if there’s any error in any of the files for it to get sorted out.
Once you update WordPress, do set the Page to Static page, and check if it fixes your issue..

NOTE: Dont worry about the custom theme.. When you update wordpress, the theme doesnt get affected. But to be on safer sides do take a backup of your theme and DB before you update 🙂