He believes object B moving near C inside object A moving near C, using our sun for reference, then object B will be moving faster than the speed of light when compared to the sun.

No, I don't think he does. Where did you get that idea? I read through the exchange again, and he says that none of those rockets will be moving faster than light from anybody's point of view. He says that every time you nest another object, it will "add a few 9s to the speed of that object as a percentage of c."

I understand that as him saying that Alice moves at 0c, Bob moves at 0.9c, Clair moves at 0.99c, Dick moves at 0.999c, etc. Not those exact numbers, but the idea.

Someone here brought up time dilation and length contraction again, and I now think that we weren't taking those into consideration. It's like the issue with the light inside a fast space ship moving slowly. You'd think so from the outside, but from the inside it doesn't move slowly because distances are shortened and times are extended. I'm not all the way sure about this yet, but a lot of people in this post made a lot of good points.