Substepping is ON, right?
In the details panel/physics of the cube there should be position solver iterations and velocity solver iterations. I always use 10x higher for both. This helps me often.
I think the carrying cube ~> pushing other cubes too hard stuff is related to the way you carry it. If you apply a sudden position shift, then this might happen without a physical force. So the cube just occupies the position of the other cube in one frame or there is instant overlap and the other cube flies of. The proper way to carry it, even if cumbersome to do, might be to the apply forces to it. For example a fixed force upwards that cancels gravity, and then two horizontal forces for left/right and forward/back. The amount of force can be coupled to the character movement. Like a hoover-cube, but with very direct reaction to movement of the character. This would also replicate real carrying in terms of forces.