Frankly, pees isn't too much of a worry, it does come from a sterile place. poop would be a concern :)
I would be wary of putting in chlorine...when you figure a jug of pool rated chlorine is used for a huge swimming pool. It isn't worth the risk of burning sensative skin :)
Stuff will get into the kiddy pool whether it is tread in, blown in, or flown in. If your child does NOT have access to it, you could cover it with a tarp. This makes me uncomfortable because I know little ones have a way of finding their way to water. However, you know your environment better that I ;)
Quite frankly, if you are keeping the kiddy pool in a sunny location, just dump it and refill it as needed. If you fill it in the morning, it should be ready by afternoon with sunlight, depending on the size of the kiddy pool.