Only the first one has comments. The red nucleus level of motor integration reflects what the intact midbrain, pons, medulla, and spinal cord can do when the corticospinal tracts have been disconnected in the cerebral hemisphere.
Many quadrupeds have no corticospinal tract, or only a rudimentary one.
In these cases, all voluntary locomotor behavior is organized by the red nucleus. Corticorubral pathways tell the red nucleus to turn this behavior on or off, but the red nucleus organizes it all. Dogs and cats do have a corticospinal tract, but apparently it functions primarily when the limbs, especially the forelimbs, are used to grasp and manipulate objects, such as eviscerating prey, or shredding nice upholstery. Their ordinary walking is red nucleus walking. Remarkably, this is true even for ground-walking primates such as rhesus monkeys.