Is the baby sick?

Well I think there is not much you can do if she has too many dirty diapers unless there is blood in it or it's lasted for a few days. I know the only time my son got hospitalized for diarrhea was when he had it for a few days, then he didn't want to drink fluids because he would go right away, then I started to see blood in his stool. The doctor's told me that if they got something in their system you want them to expel it, you shouldn't give them something to stop it because you might create more problems. Just keep an eye on her, you know fevers, blood in stool, wanting to stop drinking fluids....

It is probably just her teeth. My daughter didn't really have any trouble with her teeth till her molars came in. They get a molar at around 12 months so that probably is what it is. As long as she doesn't have diarrhea for more than 2 days she is probably fine. Even then, if she is getting fluids and no fever then she is fine. Especially since she is playing like normal.

This sounds like teething to me. My son didn't do it with his first few teeth, but around 13 months old, when his molars started trying to come in, he most certainly did.

And, I caution you that if it is what I call the teething runs, the consistency and smell of it is like nothing that you've ever experienced. And it can go on for quite a while as teeth are slow to come in. Keep an eye on her and if she develops a fever or quits wanting to take in liquids, call your doctor.

Some old women just say that if a baby eats too much she might act like that . And some say that the baby might get cold for her stomache when sleep at night . There are a lot of reasons. It might be the one you mentioned -ate sth that bothered her.
i have no idea about the tooth issue . Will the baby have stomache problem when teeth come out? My 5 month old boy is ready for his first tooth, what will happen to him?

I don't think you have any reason to worry. Most likely just a reaction to the teething