OK so you have custody. that is good. You can't expect to fit 6 years of teaching discipline and building trust into 7 months.
I would recommend just trying to connect. perhaps a few more months of bonding and consistancy will help him settle down. it certainly won't make things worse.
If it were me in your shoes I would keep him within arms reach when he was home. let him help you in everything. This will help on two levels. You will abel to bond with him, catch him doing good, make him feel important, give you an oppritunity to patiently teach him. Positive reinforcement and positive attitude. Secondly it will enable you to correct unwanted behaviors immediantly. The minute they take hold in his head. You will have a lot less need to correct and behavior won't have the oppritunity to get out of hand.
As for behavior grade I wouldn't punish for a bad grade. Just use it as foder for conversation while he is helping you tidy the house and make supper. "ahhh, a c-. Why did you get that grade? How could you have handled it better? how do you think it made so and so feel? " let him take ownership of his behavior and try not to be too personally invested in the grade. ask him how he can improve, help him set goals etc. Sinec he has had such a hard 6 years I would go easy on him for lying. Let him know he doesn't have to be scared of the truth. that an off day is a chance to learn better not to be punished. he has a lot of healing to do. I would definitely cut him some slack while he is adjusting to a new home and life he has never known.
I am generally opposed to sending kids to thier room. I tend to bring mine closer when they are misbehaving. even if it means they are literally hip to hip. I can usually find something to keep them busy though
. It has been my experiance that most kids respond very positivly to that sort of attention.