I agree with Tucker and Mothorchidgirl;
I raise Cats and neither of the two are really big enough to divide. A healthy happy cat will have shoots everywhere and, as long as you can keep it in a large enough pot, there really is no need to divide unless they are just too big and sprawling for your likes. I have seen Cattleyas that are 4 foot across with 30 or more leaves; the plant was blooming it's head off and made a phenom centerpiece orchid.
So, no don't divide. As far as the newly repotted one with the roots that were damaged. When I repot a Cat, or any orchid for that matter, I have to think about it's ability to absorb water. The roots, if not rotted or severly damaged, are still traumatized and will not absorb water immediately. I wet my potting medium, repot the orchid and then do not water the medium for two or more weeks. I simply mist the plant and allow the roots to repair and adjust to the new medium. It will take two to three weeks before they can do this. A good indicator is new green root tips forming or new roots coming out of the root stem. Until it is actively growing again, the roots are just going to rot if the medium is too wet.
With Cats I use pure coco, or coco mixes with some sphagnum. They really need air around their roots so heavy mixes are deadly to them. When I repot I always tease all the old medium out, no matter how healthy looking, and replace. You never know what is lurking in even new-looking medium.
The psuedobulb and leaves have lots of moisture, so the period of "dry feet" really does not hurt them, and it is my theory that withholding direct moisture from the roots, but providing humidity instead, "urges" the plant to put out new root growth in order to survive. It works for me.
Hope this helps - Have a great New Year!
Steve