For the Focus Finishers, hit L2 when you see the blue flare, then spam square and/or triangle on an enemy, it will ALWAYS be random, so getting 10 different ones may or may not happen.
For beat downs, when an enemy is on their knees, L2 for Focus and spam square and then you *should* perform a series of punches, at least that's what popped the trophy for me.
For Environmental Takedowns, I have no earthly idea. I saw a video on Youtube (there's literally only one single video on this trophy lmao) and they didn't even explain how to do it.
I give up on this game though. I completed NG+ for the sake of TRYING to understand more of the story, boy am I glad they added a DLC trophy for it. This shit should've stood quiet. It's so bad it's hilarious.