Yes, this bolt carrier group will work perfectly fine with standard AR-15 in 5.56/.300BLK.
This bolt carrier group (auto version) is for your AR15, M16, or M4; 5.56 NATO.
This Bravo Company MFG Bcm Bolt Carrier Group is for AR15s chambered in 5.56/.223. To use it in an AR chambered in 6.8SPC you will need a new bolt.
Yes, the Bravo Company MFG Bcm Bolt Carrier Group will be compatible with any AR15 platform rifle chambered in 5.56, .223, 300 Blackout etc.
A full auto BCG is not required for use with a Franklin Armory Binary trigger, as there is no full auto sear. However, this Bravo Company MFG Bcm Bolt Carrier Group would still fit and function as intended in your DB15.
Any 5.56 BCG should work with 300 BO, so YES.