I don't know if the dog would be considered the cause of a bacteria infection. Dogs do carry various parasites though that can spread to humans and cause a variety of health issues.
The bacteria infection is probably just that - bacteria! Bacteria can be picked up anywhere such as bathrooms, food, air, etc. Usually our body does a fine job destroying the bacteria, when for some reason our body is overwhelmed with other issues or our immunity is low then we are susceptible to illness.
The white blood cells are elevated because the child's immune system is doing it's job and fighting the bacteria.
It is always normal for a 1 year old to put everything in it's mouth.