Hi!
1- Don't know... high is... you know, high
2- HQ stacks don't alwasy assist, but they have more probabilities than corps stacks in doing it (a bonus). Still thay can fail to assist.
In the case you explains, only the units on the HQ stack will get "across the river attack" penalties.
3- Success percentage on assisting other cops/HQ in battle depends of leader strategic rating adn of distance/time to travel to the adjacent battles. So its much more probable between two areas connected by rail road.
A main river is an obstacle (increase travel time between two areas) so it will not help at all. But it also depens on the connction across the river (bridge, ferry, nothing...)
The important data to consider here is the days to travel between zones in normal circusntances (no battle assisting). The bigger the harder to get a corps to assist another. For example, mud is very bad in this regards.
Don't count on corps assisting each other on different sides of the Missisippi, but for example, between Washington and Alexandria is quite common (rail connection)
Sorry, but don't know exact numerical data of any of this
Regards!