It depends on what distance the child is running. Usually the children mile is always last so that would start at 12:30PM. This stuff is all spelled out in the event guide for the race weekend.
I am not sure your sister in law will have enough time to get her child to a race. You would have to pick up the childs bib that day as well before the race.
There are buses from all of the resorts to the expo at ESPN. There are NO buses from ESPN to the parks.
After the finish there are no buses to the parks. If the course is the same as last year the finish is at Epcot so you could walk into Epcot. Although I would think she would want to go back and change and shower rather than spending the entire day in race clothes.