Does a new hard drive contain random noise or all zeroed-out bits?

Disk drives are always low-level formatted at the factory. The bit pattern used might not be zero, but it certainly isn't random. See Wikipedia.

My understanding is that a new out of box hard drive should have all zeros. However, during all the packaging, transportation, testing, etc. that each new HDD experiences it is not out of the realm of possibility that a new drive might have some "1"s thrown in.

Ultimately, as much fun as this is to speculate, it really shouldn't matter, I doubt you would be able to identify a manufacturer that way and as soon as you write data even once the initial "signature" of the drive would be gone.

Interesting question though.