Is my unvaccinated child at risk by being near a recently vaccinated child?

Almost Never. Unless your child has an immune problem, they are at no risk. There is a much higher risk to contract the actual illness if they are around other unvaccinated children.
No. The viruses in vaccines, even the live viruses, are not communicable under all but the most extreme of circumstances. Your child is at risk of being around any other nonvaccinated child. If your child should get measles she could die. Your child would be much safer being vaccinated.
It depends. In most cases, a recently vaccinated person poses no risk to an unvaccinated person. However, in the case of live-virus vaccinations, the vaccinated person can shed live virus particles and infect unvaccinated people. The best course of action is to simply make sure that your child is caught up on their vaccinations.
No. But your kid is at risk if ever exposed to an unvaccinated person recently arrived from a foreign country who may have traveled through your local mall.Many infectious diseases are most contagious before the host realizes they are sick.We lost 40 infants to whooping cough last year but none in the 90's when they were usually vaccinated. Hope your kid does not suffer.
Important Question. 2 cases in the literature (out of millions receiving the vaccine) described for varicella: mother to 2 children and 12mo child to pregnant mother. The index cases had vesicles. More importantly- why is your child not vaccinated? If one is talking about an inactivated vaccine, then there is no risk. If its a live, attenuated virus and your child has an immunofecificency, then the risk is low.