Can children still catch chicken pox after they have been vaccinated?

Yes. Vaccines have decreased illness rates dramatically, but there are some people who don't respond to vaccines and are still susceptible. This is why multiple doses of each vaccine are needed (ie boosters). It is also a reason that a large percentage of the population needs to be immunized to make a dent in the transmission of a disease. Herd immunity protects if the immunizations aren't effective.
Chicken pox. Yes this is possible as the vaccine does not have a 100% success rate.
Yes. Data suggest 85% are protected with the first dose and over 95% after their 2nd. Usually given at 1 and 4-5 but could be given a month apart. Those that do seem to have less total spots and the spots are often atypical in character and location.