Is it true that pigeon-toed children are more likely to be good runners than children who are not pigeon-toed?

Pigeon toe. Yes. People with pigeon-toes are usually better runners since they usually have a higher arch.
Intoeing gait. This position can also lead to ankle sprains. See your podiatrist for an evaluation - prescription orthotics may be indicated to ensure proper positioning of the foot in athletics.
Theoretically yes. Intoeing tends to put the foot in a more rigid alignment which allow more force to be transmitted with push off. So theoretically, these kids have a mechanical advantage when running. However in order to run fast they still need to have some baseline athletic tendencies.
Nope. These children may instead be a little clumsy and prone to tripping as they grow older and try to do more running. See a doctor.
No. That is a myth. Every child is different, although most are pigeon-toed for the first few years. If a child becomes more pigeon-toed over time, they should be examined.