Can only Format Hard Drive as FAT

My Macbook Pro was starting to run slow out of the blue so I decided I wanted to format the Hard Drive and install El Capitan. After not being able to log into my mac, I installed El Capitan on a usb flash drive so I would be able to backup my files. When doing this, however, the main Hard Drive would constantly dismount after a certain interval of time or not show up at all when logging in. After I copied all the files, I tried to format the Hard Drive as HFS Journaled and I would get an error message. I would only not get error messages when formatting as FAT. This would even happen if I tried to separate the Hard Drive with partitions. I took it to mac support and they ran a basic diagnosis and said everything looked fine with the hard drive, but I still think there is a problem with it. I've attached photos to show the error message I am receiving. Do you think the hard drive is corrupt and needs to be replaced?