Is it possible to build a fortress without any cats? The Catsplosion problem is well known.
I mean not only build a fortress, but build a successful fortress. I know that cats helps with vermin and moods.
Do anyone has experience with 0 cats? :)
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.
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)