Hmm, everything you described sounds like normal family integration to me. I also have a cat that bites, more of a nip, and had to laugh on you comment of
I have become aware of the signs of impending bites and am much quicker now
Sampson can get a bit twitchy, and know he is going to nip me if I continue to rub him like that, and if I get in between Sam and Rufio, while they are playing, well yeah, likely I'm going to get a few punctures in my calf.
Currently, my F2 twins have started to assert their authority over the household. They are a bonded commando force of nature. One cries fur help and the other is there as backup.
Recognizing what each cat is trying to tell you, is the best way of not getting blooded .

and then sometimes there is the picking up and object that is fluffy, or looks like toilet paper, well, that's just bad timing, ouch, that hurts type thing.
Have found the best way fur one not to have their toes attacked, is not to wiggle them, move your feet under the sheets, and yeah, one is just asking fur it.
Have found the best method of learning gentle, is when it hurts, I'll cry, and say, why did you hurts daddy, boohoo, be gentle, that hurts, you made daddy bleed, gentle, gentle.