Probably yes. A child who is exposed to diarrhea at daycare, will probably get sick because kids at daycare tend to pass germs to one another every day. If the child has protection against the diarrhea, either from having a vaccine (against that exact germ), or from having had the diarrhea illness (caused by the same exact germ) in the past, then the child may get just a mild illness or no illness at all.
Quite possibly! Most forms of diarrhea in infants and young children are contagious.Although being exposed does not automatically mean your child will become sick, there'a a good chance he/she will. Make sure your daycare provider uses excellent handwashing and disinfecting procedures, esp around diaper changing.
Probably! The vomiting and diarrhea illnesses of the winter are mostly viral and are very contagious. In fact they can live on surfaces for up to six months! gross, huh? There is a good chance your child will get it. Once the exposure has happened there is not much you can do. Expect a 24 to 72 hour incubation period, and cross your fingers. Keep washing your hands, because you may get it too!