Copper I am so sorry you are going through this. Can you get post adoption support where you are?
My birth dd is 10 now and went through a small amount of what you describe. My adopted ds is now 4 (came to us at 3) and does some of the things but no where near as much.
We now do take a break, which I can join him in if necessary. This is cushions in a safe space or the bottom of the stairs if he is safe to sit there (e.g. not go up them and jump down etc). It is no toy zone for about 2 minutes and a time to calm down. We do not demand an apology generally but i would for hurting another. It is not a punishment but a time to cool down. For us it does work sometimes. Not always. Both my kids need a cuddle when upset but the timing is crucial! Ds needs it quite soon after but dd needs time (usually in her bedroom) to calm down before the cuddle.
Although time out/take a break or whatever is not specifically a punishment if our children do not go on the spot they get a punishment like loss of a toy for a few minutes. Although these punishments and sanctions will not 'cure' the bad behaviour I do feel it is useful to get children used to them because when they are at school behaviour like this will not be tolerated by teachers so the kids need to know it is anti-social. My son went through a bit of a biting time, always at his sister, never at school. This behaviour may not manifest at school or pre school, it could be just in the family home, like mine is. I am usually the target of both kids if they feel angry, not at the same time, my husband does not get hit etc ever. For this reason I feel my kids can control it a bit and are choosing me because either know I will always love them (their dad would too but maybe women seem to be more the target for kids aggression).
I am going to look into this book someone mentioned, 'Out of Sync child' as I think my dd has some sensory processing issues. I feel my dd may have attention deficit disorder or possibly ASD. But her behaviour manifested at 5 when she went into Year 1 at school, so may be different to your child's. Also my dd is not adopted.
I do feel some things they do grow out of and work through, being able to talk more and say how they feel, what they need etc might help your child.
The behaviour might be testing you, as others have suggested.
Please do keep yourself safe. I know it is hard. I have had some degree of it with dd and have ended up being strict on this one area, I make sure she knows I love her unconditionally but I won't be attacked and I have to be very careful how I protect myself as sometimes protective moves can look aggressive to child (e.g. raising your arm to protect your face) but as a mum you must be protected, IMHO.
I think you need more advice and help, I hope you can get it in your area, our area post adoption team has been brilliant. Please do not wait for it to pass, just see what is available.