ALL hard drives will die.
Yours is dying. The bad sectors are not necessarily a symptom, but the noise is a warning. Many new drives ship with bad sectors.
Formatting will not address bad sectors.
The DVD cannot, in and of itself, cause bad sectors or corrupt files. I/O errors can cause corrupted files but do not cause bad sectors. Bad sectors can cause I/O errors.
A perfectly good drive will experience Bit Errors. The smaller the drive, the less likely a bit error will occur on any single transaction, but it is still possible.
Think of your HDD as an old-fashioned vinyl LP. Playing the LP causes wear and tear. Same with the HDD.