"What exactly does a cat do besides eat, sleep and eliminate??

Liz the Squeak - "A cat keeps your home free of rodents, spiders and milk spills, whilst performing a stimulating and decorative function as a throw pillow."

LOL, Liz. And I love that "throw pillow" line. Cats also serve as vibrating hot water bottles.
(I have yet to meet a dog that can purr.) Plus they perform all sorts of entertaining acrobat and clown tricks, usually with no training whatsoever.

This is not to diss dogs. They have their value, too, as mentioned above. However, they are much higher-maintenance companions (including often smelling really foul unless bathed regularly by their humans), and this makes them less suitable as animal companions for many people.   Having a dog is a bit like having a kindergarten-age kid.   They may not be all that destructive to your house and home, but they cannot be left long on their own, for their own safety and health.   Cats, on the other hand, are more like teenagers.   Much more independent and less likely to starve, mope, or need help with their daily routine - but also more likely to trash the furniture when you're not there.