Here is the rule on Right of Way and Yielding in Minnesota for funeral processions from the state’s drivers manual (Boy, does this bring back memories of going through this!):
“When a funeral procession identifies itself through use of headlights or hazard warning lights, you must yield to the entire procession.”

I had a quick Google around and opposing traffic is free to travel with regional etiquette sometimes stating it is a sign of respect to the deceased to pull over (Southern US) - but it is not required. Given the size of populations now I would imagine this is a vanishing custom unless we are speaking of a small isolated town where everyone knows everyone else. (id est 1,000 people or so)
Cross traffic right of ways on the other hand have pretty common rules nationwide that in short state - the group of cars travels as a single unit so if it takes too long to get through a light then opposing traffic must wait and treat the light as still red to them. Since the group is a single unit you do not cut into it nor cross it. Think of it as a really long single vehicle.