how do you groom your yorkie?

all this will depend on the amount of mats.....but a very popular and semi-easy to care for style for this type of dog is a "westie clip" That is what we call it, the name may vary in other locations, but what it is is a short back with the legs left longer and a skirt hanging down. This gives the illusion of length without all the intensive brushing. It takes only a moderate amount of brushing. Your other options are on either side of the spectrum from here....either entirely short (the very easiest to maintain) or all long hair all over (the most difficult to maintain). Keep in mind that the more hair the more brushing at home AND the more trips to the groomer.
Sweaters are cute and good for a dog with short hair, but if you use a sweater and keep the hair long, prepare for extra brushing. First, the static will cause the hair to stick together and second, just the tightness of the sweater itself will cause the hair to mat. Once the hair is tangles a mat forms in the blink of an eye, and there is only so much that can be combed out before you need to shave it off.
Use your dogs temperament and hari texture to determine the type of clip. A dog that hates brushing will never be happy if you insist it has long hair! And some dogs DO NOT have the proper texture for certain clips. (just like we all can't be super models).

Now for the cat/dog relationship......I yield the floor to someone with more expertise than me.

(sorry so long winded)