To answer your questions -
Rubber flooring is pet friendly and moisture resistant making it safe for both your kitchen and bathroom. Since you are considering rubber tiles, you may have to worry about urine seeping to the seams. I recommend rolled rubber as there will be fewer seams. A glue down installation is generally the best choice as it will bond to the subfloor and not allow any liquids to flow under the seams in case your cats have an accident.
What is your exisiting floor made of? Usually it is best to remove the exisiting flooring as it can make installation more difficult being that you may need to cut the bottom of your doors for the flooring to fit underneath.
When I know what flooring you have now, I can give you a more complete answer, but I can tell you that your subfloor must be level and free of and bump or imperfections as these can show through the rubber flooring over time.
I have cats too so I have dealt with the frustration that they can cause when it comes to flooring. In their room where the litter box is, I laid vinyl flooring recently. It's also pet friendly and easy to clean. My cats are weird about their cat litter. If I don't get them tidy scoop for multi cats they will start acting up so you might want to try switching brands of cat litter. I also have one of those cat litter boxes that you can just roll to clean and pull out a drawer to empty the clumps. I highly recommend it!