I don't know about slingshots, given the lack of truly elastic materials being available and those that were available probably were prohibitively expensive to make such things common, let along making them a simple weapon.

As for your normal slings, I could see that as being functionally sound, except for the 2 squares of space to fire, as we know that people certainly don't take up all the space in a 5 foot square and slings, given they could be used as a headband or wristband, don't need 10 feet of space to operate in. Historically, slingers could be put into close order with each other to make a rain of stones or lead, much like bowmen can, so they certainly didn't need that much space.

The range seems closer to historically accurate, given the distance you could cover with a launched bullet from one. Also making them exotic makes sense as well, given the famous slingers of old had years of practice from childhood on to get the lethal accuracy that they were known for.

Finally, I'm not sure about normal stones as ammo dropping the range and damage so dramatically. They certainly weren't as aerodynamic or dense as a lead bullet but performance didn't drop THAT much. I'd drop damage one size category, -1 to attack and maybe drop range 10'-20', certainly not in half.

Them's my 2 coppers. Take as you will.