Your SD card most likely got broken: I've had the exact same issue a while back on another device. I'd recommend using a card reader to attach the card to your computer, and do a check for bad blocks. In my case it turned out almost all sectors were "bad" – with the effect that the old data remained, and all "writes" went to Nirvana:
- delete file: reappeared after boot
- add file: disappeared after boot
Reason is, those changes are just kept in your device's cache, so it appeared they were performed correctly. On boot, that cache was purged.
So your solution would be: Get yourself a new SD card. Optionally copy over the files which you want to keep from your old one. Now everything should work fine again.