Thank you Michele for some very good questions. The activities I choose to do are based on how receptive and able I feel the child is. Many of my children are very young (between 18 months and 3 years) and a big part of therapy is having them separate from their parents. Some are afraid of the horse and others take to it right away. The older kids (over 3 years of age) are usually more willing to try more challenging activities such as standing on top of the horse more quickly, but it all depends. One almost 3 year old is terrified of lying supine on the horse but will stand on top and play with my toys.
Parents have told me that various activities have improved sitting and standing balance and gait. I often see improved speech and visual attention than they might have otherwise and I have been told that this is the only time they speak.
I have also been more daring over the years.When I started I was terrified of a child falling off and barely let them move and now I push and push. So it comes with time and experience.As far as carryover goes- I sometimes recommend that the parents do specific fine motor activitiesat home and practice holding some positions such as kneeling and quadruped on land to make them more familiar and receptive to holding these positions while on the horse. 
I can't say how much carryover there is but it is certainly good for the child and the paretn to know that they have these abilities while on the horse and that they can strive toward carryover at home.