you got it confused.
although the practice is not done much anymore, there are still available utilities to low level format (try google) - should the situation deem necessary - such as this case in this thread.
as with a new drive purchase. there is a level of trust toward a manufacturer. hence no need for low level format. as with a used drive purchase. you definitely want to low level before putting your precious data on it. some of us will low level format a new drive just for our personal peace of mind.
low level format is basically zero-ing out the drive. while the platter is being zero out, it also gets checked in the process. a bad spot will not zero out and is marked bad.
once the drive get zero out. the next step is partition setup.
after that. then the high level format (aka quick format). which is the allocation table (FAT/FAT32/NTFS/etc).
now the drive is ready for data. all new data written will have a fresh magnetic signal.
No. True low level formatting puts the drive tracking and sector positioning data on the drive. In order to do this on any modern drive, you would need the head alignment / magnetic tracking system at the factory to lay down the cylinder tracking patterns. There is a sticker covered hole near the head servo assembly where these machines connect and externally position the heads and has the heads write the pattern.
Zeroing out a drive is not a low level format, it is just Zeroing. The check is just disk checking. You can do destructive testing if you want where you write and read back test patterns.
There is no way in the field to rewrite the positional data on the drive. It used to be that one side of a platter was reserved for this and the write head was disabled after alignment. This may or may not still be true. It was the main reason why 2 platters were needed for 2TB drives using 1TB platters etc.