I too have a daughter that just turned three this week. I'm not sure at her age that I can impress respect for animals on her, at least not that she'll always remember. We have a very patient dog who will put up with just about anything, and we have to tell our daughter over and over to "be gentle." She does seem to remember most of the time, but when she gets excited, she may just run up and grab the dog, or pinch her, or try to whack her with something. I think at this age, we just don't have much choice but to watch them all the time with animals.
Maybe you and the cat could take a break every so often if you put the cat in another area where your daughter can't get to her. Or how about a kitty tree or window seat where she can jump up out of reach when she wants?
One thing you might want to think really carefully about is the declawing. I used to be a vet tech, and his (the vet's)first priority when someone came in to inquire about having their cat declawed was to try to talk them out of it. Now, if they absolutely insisted, he would do the surgery, because he would rather do that than to have them get rid of the cat, or take it to a vet that wasn't as caring. But he made sure that they knew that there could be serious complications. Declawing isn't exactly what it sounds like, what's actually being done is that you amputate each toe at the first joint. Like cutting off each of your fingertips at the first joint. There can be problems with anesthesia, with infection and sometimes even with the way they walk. If they get outside, they have no protection, and we saw lots of cats each week that got out for a couple of hours, and came back ripped up. Now, years before I was a tech, I had a cat declawed, all four feet, I was traveling on the road, and she was tearing the furniture up in every hotel room. Her feet got infected, and washing her feet while squeezing the pus out was no fun for either one of us. I swore I would never do that again, and with all my other cats I've trained them to sit still in my lap while I trimmed their claws very short. If you'd like to send me a message through Mamasource, I'll tell you how I did it.
One thing that's really important, make sure you get that cat spayed or neutered (if it's not all ready)! A female will do whatever she has to to get out when she's in heat, and you will eventually get surprised with a big batch of kittens. Their cycles can start as early as their fourth or fifth month, and can repeat every two to three weeks. Cats can seem like they're always in heat. Males, on the other hand, will fight and spray everything in sight. I can't tell you how many incidents we had with un-neutered male cats. If you wait until they mature to neuter them, and they've started spraying, you may not be able to get them to stop.
I hope things work out for you. I can't imagine not having animals around, and I think they can be really good for children. I let my kids take turns feeding the dog each day, they love it and it seems to help them understand that they need to take care of her. And maybe the cat will grow on you, you never know. It can't be any fun to be afraid of animals, especially when they seems to be everywhere these days. I've seen people in Sarasota take their dogs into just about every store, and now I think they can even take them to lunch at the Ritz-Carlton!