Page 58, Chronopia, War in the Eternal Realm, Rulebook:
"If a model Choose to Run, the only other Actions it may take during that activation are additional Run or Move actions."
The Book is Clear, you may NOT Run and Shoot during the same Turn. I think you declare your Run, before Activating the Model. Otherwise, you may: 1 Action: Shoot, then, with the second, Run.
Page 60, Aim:
"Once you have Aimed a target, you continue to get the Bonus for Aiming, until you Move, your Target Moves, or Your Turn ends."
Because "Move and Shoot" is a Move Action, you can not Aim and Move or Move and take a Aimed Shoot when moving with a Mounted Archer. Ok, i think you may Aim and Move, but it is ineffectiv, because you lose your Aim Status, at the same time (because you´re moving...)
I hope i can help