Can a vaccinated child be a chicken pox "carrier" and give it to an unvaccinated child or adult?

No. Even though the vaccine is a live virus you cannot spread it to others. However people immune suppressed may be ate some risk to getting the disease.
No, but... According to cdc, vaccinated children cannot transmit the vaccine virus, or at least it has never been documented. However, when the vaccinated child develops a rash following vaccination, that's in essence a mild case of chicken pox, and that kid is contagious until the blisters form scabs. But as far as we know, there is no such thing as a "carrier state" for vaccinated kids (or adults).

Related Questions

Is it possible for a vaccinated child to be carrying chicken pox and give it to an unvaccinated person?

Very rarely. You can only be contagious with chicken pox if you actually have the infection. People vaccinated against chicken pox have about a 90% chance of being immune. If someone was vaccinated and didn't become immune, that person could get infected with the chicken pox and then pass it on. Read more...
Both are possible. Because vaccines are not 100% effective, occasionally a vaccinated child gets a case of real chicken pox (not from the vaccine) and could pass it to an unvaccinated, non-immune person. Once in a while, a child who got the vaccine a few days ago can have some skin bumps containing the vaccine chicken pox virus, and could pass that virus to a non-immune person, who would be getting a "free vaccine". Read more...