Yes to 1,2,3. If you sign a contract you can get the Dish or DirecTV hardware at no cost but the antenna equipment required to receive the programming in an RV is on us.
#4, yes, the Hopper can be used but not with any of the dome type antennas. You would need a full sized open face antenna that you mount on a tripod and point at each new location or an automatic roof mounted antenna like the Winegard Trav'ler.
With DirecTV you can use their Genie with a dome but you would not get HD. So, if you must go with an open face antenna, and you don't mind a contract, I would go with the Genie and DirecTV.