Don't really see much of a problem here.
On the first ball he should be going for it with his left foot and just looking to knock it out of play, because it is an awkward ball to deal with.
On the second ball, Bonucci is quite rightly shaping his body to be ready to move with Maxi Lopez, so he was inevitably going to struggle when the ball went to the other side of him.
Sure, it looks pretty clumsy, but he's really isn't doing much wrong.
Well I think you actually listed a problem. First, he's trying to bring the ball down with 3 players within a couple yards of him, 2 of them being Chievo strikers. Of course, he does it with his right foot which gets him screwed into the ground. From there he can't turn. You have to keep your body open at all times so you can remain mobile, and especially as a CB you never want to have your face to your own goal. So you have to use your left foot there.
And from there, he has no idea where Ogbonna is. He's right behind him, ready to deal with the bouncing ball. Instead, he's flustered as usual and sticks out his heel intending to clear it 50 yards I guess? Predictably it goes 5 centimetres and lands on the striker's foot. To stick out your leg like that is a joke. The ball is not going anywhere even if you connect. You're better off to forget the ball and shield the attacker, something Chiellini would do, and try to open your body towards Caceres. He's not shaping his body for anything, his hips are turned almost completely up the field, they're not turned to run with anyone.
His instincts and athleticism get him in awful, awful positions where from there you just throw up your hands and say, "I guess that's all he could do".
FFS, he played a delicate one-two with Schelotto.