The Presma .308 Barrel Vise Block Rod with Torque Block for .936" barrels is not designed for .300 Blackout barrels by default, unless your specific .300 BLK barrel has a .936" gas block journal diameter.
No, the other one will work with.
https://www.opticsplanet.com/presma-308-barrel-vise-block-rod-w-torque-block-for-936.html
Yes, it works.
The Presma AR-15 .223 Barrel Vise Block Rod w/ Torque Block could still be used for installing a barrel nut even if you do not use a .750 gas block.
Yes, this will work for 556 as well.
I found the information below by reading the product information sectection. For more information, Presma has their own website. My quick look at their site also said it take 70+ ft/lbs of torque. That should cover most installations.
Specifications for Presma AR-15 .223 Barrel Vise Block Rod w/ Torque Block for .750 Barrels:
Finish: Anodized
Fabric/Material: 7075-T6 Aluminum, 6061-T6 Aluminum
Quantity: 1
Caliber: .223 Caliber
Gun Model: AR-15
Gun Type: Rifle
Length: 11 in