It's won't crystallize into ice unless there's some transition energy provided, no.
One does not "provide" energy to water to freeze it. A cold environment does the opposite by removing kinetic molecular energy.
You are right but for the wrong reason. The corollary question is "Does ice melt at 0 degrees Celsius?"
Both cannot happen at the same temperature. In fact, neither one can. The environment has to be warmer than 0 to melt ice and colder than 0 to freeze water. At precisely zero, there is perfect equilibrium, as in the triple point. This is overlooked by all chemistry classes and professors as they teach the zero mantra.