I know you want to show your home at every opportunity, but if your kids are napping, then I would tell the realtor that the house isn't available to be shown at that time. In fact, if naptime is ALWAYS at the same times, you could make it part of the info... "call for appt... not available for showing between 11:00-2:00" or whatever naptime is.
I would not allow the house to be shown with your kids and a babysitter there. That is a very awkward scenario for your sitter. Just tell the realtor that your home isn't available at that particular time. If they want to see it, they will figure out an alternative time... often the agent can flip-flop the list of homes, starting at the other end of their list and your home can be shown an hour or two earlier or an hour or two later than first requested.
Honestly, people are uncomfortable looking at your house with you there. They want to be able to open closets and cabinets and look around in the garage without feeling weird about it. Your presence will be inhibiting to them and affect their "feelings" about your house as a potential purchase.
As for the cat, stick it in a cat carrier and take it with you when you leave for the showings. I did this with our dog when we were selling our house a few years ago. Every time they called wanting to show the house, I loaded up the kids (4 yrs and 7 yrs) and the dog (1 yr old- so still really a puppy) into the car and drove around, or took them to the park for an hour. It was really annoying when the showing ran late (or they ended up being no-shows)... but it happens, so be ready to experience it and try not to let it get to you.