A: No. The install data (under Game Data Utility) is separate from Saved Data Utility. You can safely delete it to free up space; your progress remains.
A: Possibly. You can remove the hard drive, connect it to a PC via a SATA-to-USB adapter, and use software like HDD Raw Copy or PS3 HDD Reader (requires encryption keys). It’s advanced, but professional data recovery services exist for PS3 HDDs.
Right now, power on your PS3, navigate to Saved Data Utility, and copy your God of War 3 save to a USB drive. Do it before the next power flicker, the next hard drive stutter, or the inevitable march of time. Kratos would approve. Keywords integrated: God of War 3 save data PS3, backup God of War 3 PS3, transfer save files PS3, PS3 savedata utility, God of War 3 corrupted save, PS3 USB save transfer, God of War 3 trophies.