Dwarf fortress without cats

Yes, it is possible to have a successful fortress without any cats. The only downside with not having cats is the lack of vermin hunters, but vermin are a negligible nuisance, as all they do is slightly lower the dwarves' mood and deplete food stocks.

The mood penalty vermin inflict upon dwarves will be offset a hundred times over by a grand dining room. If you're concerned about food stocks (you shouldn't be, vermin don't consume anywhere near as much food as the bearded bipedal vermin that infest your fortress), you can make bins out of metal to stop the vermin getting at the contents.

You can also keep a bunch of male cats and kill all female cats (if you do the reverse, all female cats will be pollinated when a single male cat is brought by a migrant).

Here's what DF wiki has to say on ways to get rid of cats that have already adopted a host dwarf (cats that haven't adopted a dwarf can be simply butchered)

• Execution Traps. A deep drop into magma, water, or a very hard floor will usually do the job.
• Since the introduction of pen/pasture zones a cat owner will happily place their cat in the pen/pasture, pull the adjacent lever and walk away while the soft splat of a kitten's cruel end is lost in a sound of thunder.
• Using a dump zone to throw items off cliffs or buildings and pasturing cats below it will often seriously injure cats - especially if the drop is at least 20 z-levels. Cobaltite, cinnabar, and other heavy materials usually work best.
• As cats appear to be attracted to vermin inside a food stockpile, putting a pet impassable door in the entrance of the food stockpile and constructing an upright spike on either side of this door and then connecting a lever to these spikes can kill the already "pet" cats in a relatively safe manner. NOTE: Be sure to check the door for dwarves before pulling the lever or they might be harmed. A really fun way to kill cats is to put a pit over a volcano or other deadly fall and simply assign a kitten to the pit. A group of unsuspecting dwarves will drag your cuddly pets and drop them straight into the mouth of the volcano, or even your very own moat! With magma, you get to see a large flare of smoke to assure the destruction of your cuddly friends. sadistic laughs
• Modding. Removing the creature token [ADOPTS_OWNER] will, yes, stop cats from adopting owners. If you're feeling vindictive, however, you can instead remove one or more of the cats vital organs, causing them all to cease being alive. Note that this solution will most likely be the start of a tantrum spiral.

If you can get them (usually by trading with elves), Peregrin Falcons make an excellent cat replacement. From the wiki article, Peregrin Falcon, they don't adopt owners, do hunt vermin, and as birds their births are easier to manage. They can be hard to obtain however. The article even implies that they are more single minded about hunting vermin, possibly because they don't bring their kills back to their owner.