Is keeping HDD ON the anti static paper likely to damage it?
Hard disk controller boards are surprisingly resistant to ESD damage. I've handled hard disks very rougly in the past, and stored them inside bags made of polythene, nylon and various fabrics without any damage whatsoever. Obviously there's still a potential for damage (if you'll excuse the pun), but it's certainly not something that's likely to happen by just brushing your hand over the board.

That being said, the external surface of some anti-static bags - especially those used to store consumer electronics - conducts a small amount of current. If you power the drive on whilst it's sat on the bag, it might well be the equivalent of hooking up a ~10M&ohm; resistor between arbitrary contacts on the board. I don't see this being too much of a problem on a hard disk, since everything sits quite flush to the board and few surface contacts are likely to connect with the bag's surface, but I have seen this problem with the underside of motherboards.

Either way, it sounds like you've handled it quite safely, excluding the latter part. If the disk has failed, I'd be more likely to contribute it to a sharp shock than any electronic fault.