Empire in Flames is an official expansion and have included the Town Cryer article 'Blazing Saddles' by Robert J. Walker and Roger Latham regarding mounted warriors. You have the answers to most of your questions there, and some nice skills and rules to not make mounted heroes OP. If you use that rules, there's the 'horse archer' skill, that allows you to shoot while riding a runing mount with -1 to impact, implying you may not shoot when running unless you have that skill.
Road Warden is an official hired sword, and a mounted marksman (an expensive but really good one). Both, Road Warden and Highwayman have the 'expert rider' skill, that allow them to move and shoot without negative modifiers, implying you have -1 as usually when you move and shoot while riding. That skill is not avaliable to heroes, however I'd house ruled it and make it avaliable, but not stackable with 'horse archer' because shooting and running while riding is too good and it should always give you -1 to impact.
Remember that ruleset is optional, and if you are using it you always need first the 'rider' skill for the specific creature you are mounting.
If I'm not wrong mounted should count as elevated position, but as large target also.