Do you put your cup back in the refrigerator every time you are done drinking out of it for a few mins? There's no reason then to do it to your daughter's. There's nothing wrong of course with putting it in the fridge, but I wouldn't worry too much about it. Paridoxially, worrying too much about bacteria making your child sick, can inadvertantly make your child sick. Overuse of antibacterial products has caused mutation in many illness causing bacteria, making them harder to treat, and because children are not being exposed to some bacteria in small amounts so that thier bodies can build anti-bodies prior to serious exposure - they become much sicker than they might otherwise. Their bodies have to start from scratch against a large infection instead of being able to rely on previously created "tools" to fight it. There was truth to the old mothers' saying that kids should eat a tablespoon of dirt each year. Of course I'm sure they meant collectively, not literally.. :)
I've always left my kids' cups out all day (provided they are not drinking milk, and all 6 are perfectly healthy. "They" also say that kids should have fluids available all day, and if you know anything about a busy-discovering toddler, once they get playing they're not likely to think about that sippy cup again (and your daughter of course is too young to ask). So leaving the cup out where she can see it has other benefits as well. Besides, you can rest even easier that it is harder for "backwash" to get into a sippy cup due to the valve than regular cups.
As far as leftovers, I have to agree with other posters that as long as the babyfood jar is returned to the refrigerator in a reasonable amount of time, there's nothing wrong with leftovers either.