A child has been around someone that has pink eye, so how long before that child shows eye symptoms?

Depends. Assuming the pink eye is caused by a viral infection (most likely), the time from contracting the virus to showing symptoms can be anywhere from 4-10 days. If it was a bacterial infection, symptoms usually being sooner.
Depends. The development of viral conjunctivitis depends upon exposure to an infected agent and the susceptibility of the individual exposed. It is generally felt that viral conjunctivitis will affect about 15% of those exposed, so a child so exposed is more likely not to get it. If the child is susceptible, it will show up in 3-5 days for most infections.